%------------------------------------------------------------------------------
% File     : Leo-III---1.7.7
% Problem  : GEO452+1 : TPTP v8.1.2. Released v7.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n026.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Fri May 19 11:10:44 EDT 2023

% Result   : Theorem 198.40s 131.64s
% Output   : Refutation 209.31s
% Verified : 
% SZS Type : ERROR: Analysing output (MakeTreeStats ran out of CPU time)

% Comments : 
%------------------------------------------------------------------------------
thf(p_type,type,
    p: $i > $o ).

thf(s_type,type,
    s: $i > $i > $i ).

thf(bool_type,type,
    bool: $i ).

thf(i_type,type,
    i: $i > $i > $i ).

thf(fun_type,type,
    fun: $i > $i > $i ).

thf(cart_type,type,
    cart: $i > $i > $i ).

thf(real_type,type,
    real: $i ).

thf(linear_type,type,
    linear: $i ).

thf(matrixu_vectoru_mul_type,type,
    matrixu_vectoru_mul: $i ).

thf(matrix_type,type,
    matrix: $i ).

thf(int_type,type,
    int: $i ).

thf(num_type,type,
    num: $i ).

thf(intu_pow_type,type,
    intu_pow: $i ).

thf(intu_ofu_num_type,type,
    intu_ofu_num: $i ).

thf(numeral_type,type,
    numeral: $i ).

thf(u_0_type,type,
    u_0: $i ).

thf(numu_ofu_int_type,type,
    numu_ofu_int: $i ).

thf(list_type,type,
    list: $i > $i ).

thf(map2_type,type,
    map2: $i ).

thf(f_type,type,
    f: $i ).

thf(cons_type,type,
    cons: $i ).

thf(h1_type,type,
    h1: $i ).

thf(t1_type,type,
    t1: $i ).

thf(h2_type,type,
    h2: $i ).

thf(t2_type,type,
    t2: $i ).

thf(prod_type,type,
    prod: $i > $i > $i ).

thf(zip0_type,type,
    zip0: $i ).

thf(l2_type,type,
    l2: $i ).

thf(c__type,type,
    c_: $i ).

thf(hd_type,type,
    hd: $i ).

thf(tl_type,type,
    tl: $i ).

thf(length_type,type,
    length: $i ).

thf(l0_type,type,
    l0: $i ).

thf(last0_type,type,
    last0: $i ).

thf(h0_type,type,
    h0: $i ).

thf(k_type,type,
    k: $i ).

thf(t0_type,type,
    t0: $i ).

thf(itlist2_type,type,
    itlist2: $i ).

thf(b0_type,type,
    b0: $i ).

thf(nil_type,type,
    nil: $i ).

thf(replicate_type,type,
    replicate: $i ).

thf(numu_gcd_type,type,
    numu_gcd: $i ).

thf(intu_gcd_type,type,
    intu_gcd: $i ).

thf(exp0_type,type,
    exp0: $i ).

thf(p0_type,type,
    p0: $i ).

thf(all2_type,type,
    all2: $i ).

thf(cond_type,type,
    cond: $i ).

thf(f0_type,type,
    f0: $i ).

thf(hreal_type,type,
    hreal: $i ).

thf(trealu_eq_type,type,
    trealu_eq: $i ).

thf(trealu_add_type,type,
    trealu_add: $i ).

thf(destu_real_type,type,
    destu_real: $i ).

thf(x1_type,type,
    x1: $i ).

thf(y1_type,type,
    y1: $i ).

thf(realu_add_type,type,
    realu_add: $i ).

thf(mku_real_type,type,
    mku_real: $i ).

thf(finite_image_type,type,
    finite_image: $i > $i ).

thf(finiteu_index_type,type,
    finiteu_index: $i ).

thf(destu_finiteu_image_type,type,
    destu_finiteu_image: $i ).

thf(intu_max_type,type,
    intu_max: $i ).

thf(max_type,type,
    max: $i ).

thf(trealu_mul_type,type,
    trealu_mul: $i ).

thf(realu_mul_type,type,
    realu_mul: $i ).

thf(nadd_type,type,
    nadd: $i ).

thf(destu_nadd_type,type,
    destu_nadd: $i ).

thf(naddu_mul_type,type,
    naddu_mul: $i ).

thf(mku_nadd_type,type,
    mku_nadd: $i ).

thf(intu_min_type,type,
    intu_min: $i ).

thf(min_type,type,
    min: $i ).

thf(a0_type,type,
    a0: $i ).

thf(naddu_rinv_type,type,
    naddu_rinv: $i ).

thf(div_type,type,
    div: $i ).

thf(t__type,type,
    t_: $i ).

thf(l__type,type,
    l_: $i ).

thf(p__type,type,
    p_: $i ).

thf(suc_type,type,
    suc: $i ).

thf(intu_mul_type,type,
    intu_mul: $i ).

thf(realu_le_type,type,
    realu_le: $i ).

thf(realu_ofu_num_type,type,
    realu_ofu_num: $i ).

thf(realu_lt_type,type,
    realu_lt: $i ).

thf(sqrt_type,type,
    sqrt: $i ).

thf(in_type,type,
    in: $i ).

thf(setspec_type,type,
    setspec: $i ).

thf(card_type,type,
    card: $i ).

thf(gspec_type,type,
    gspec: $i ).

thf(finite_type,type,
    finite: $i ).

thf(sum_type,type,
    sum: $i ).

thf(finrec_type,type,
    finrec: $i ).

thf(insert_type,type,
    insert: $i ).

thf(empty_type,type,
    empty: $i ).

thf(bit1_type,type,
    bit1: $i ).

thf(realu_inv_type,type,
    realu_inv: $i ).

thf(realu_abs_type,type,
    realu_abs: $i ).

thf(realu_neg_type,type,
    realu_neg: $i ).

thf(monoidal_type,type,
    monoidal: $i ).

thf(neutral_type,type,
    neutral: $i ).

thf(iterate_type,type,
    iterate: $i ).

thf(intu_lt_type,type,
    intu_lt: $i ).

thf(intu_neg_type,type,
    intu_neg: $i ).

thf(intu_le_type,type,
    intu_le: $i ).

thf(intu_divides_type,type,
    intu_divides: $i ).

thf(intu_add_type,type,
    intu_add: $i ).

thf(s0_type,type,
    s0: $i ).

thf(inf_type,type,
    inf: $i ).

thf(realu_pow_type,type,
    realu_pow: $i ).

thf(bit0_type,type,
    bit0: $i ).

thf(psubset_type,type,
    psubset: $i ).

thf(l_a__type,type,
    l_a_: $i ).

thf(m__type,type,
    m_: $i ).

thf(dimindex_type,type,
    dimindex: $i ).

thf(univ_type,type,
    univ: $i ).

thf(row_type,type,
    row: $i ).

thf(intu_sub_type,type,
    intu_sub: $i ).

thf(subset_type,type,
    subset: $i ).

thf(unions_type,type,
    unions: $i ).

thf(q__type,type,
    q_: $i ).

thf(u_match_type,type,
    u_match: $i ).

thf(h__type,type,
    h_: $i ).

thf(naddu_eq_type,type,
    naddu_eq: $i ).

thf(naddu_inv_type,type,
    naddu_inv: $i ).

thf(destu_hreal_type,type,
    destu_hreal: $i ).

thf(x_type,type,
    x: $i ).

thf(hrealu_inv_type,type,
    hrealu_inv: $i ).

thf(mku_hreal_type,type,
    mku_hreal: $i ).

thf(fact_type,type,
    fact: $i ).

thf(nsum_type,type,
    nsum: $i ).

thf(realu_div_type,type,
    realu_div: $i ).

thf(sup_type,type,
    sup: $i ).

thf(delete_type,type,
    delete: $i ).

thf(casewise_type,type,
    casewise: $i ).

thf(clauses_type,type,
    clauses: $i ).

thf(n3_type,type,
    n3: $i ).

thf(vector_type,type,
    vector: $i ).

thf(map0_type,type,
    map0: $i ).

thf(o_o__type,type,
    o_o_: $i ).

thf(d__type,type,
    d_: $i ).

thf(vsum_type,type,
    vsum: $i ).

thf(odd_type,type,
    odd: $i ).

thf(realu_gt_type,type,
    realu_gt: $i ).

thf(trealu_ofu_num_type,type,
    trealu_ofu_num: $i ).

thf(m_type,type,
    m: $i ).

thf(realu_sub_type,type,
    realu_sub: $i ).

thf(vectoru_norm_type,type,
    vectoru_norm: $i ).

thf(y_type,type,
    y: $i ).

thf(vectoru_sub_type,type,
    vectoru_sub: $i ).

thf(n20_type,type,
    n20: $i ).

thf(diff_type,type,
    diff: $i ).

thf(image_type,type,
    image: $i ).

thf(o_type,type,
    o: $i ).

thf(g_a__type,type,
    g_a_: $i ).

thf(hrealu_mul_type,type,
    hrealu_mul: $i ).

thf(hrealu_ofu_num_type,type,
    hrealu_ofu_num: $i ).

thf(matrixu_mul_type,type,
    matrixu_mul: $i ).

thf(mat_type,type,
    mat: $i ).

thf(intu_sgn_type,type,
    intu_sgn: $i ).

thf(rows_type,type,
    rows: $i ).

thf(filter0_type,type,
    filter0: $i ).

thf(r__type,type,
    r_: $i ).

thf(intu_abs_type,type,
    intu_abs: $i ).

thf(woset_type,type,
    woset: $i ).

thf(fl_type,type,
    fl: $i ).

thf(less_type,type,
    less: $i ).

thf(distance_type,type,
    distance: $i ).

thf(vectoru_add_type,type,
    vectoru_add: $i ).

thf(n_type,type,
    n: $i ).

thf(p1_type,type,
    p1: $i ).

thf(n10_type,type,
    n10: $i ).

thf(hasu_size_type,type,
    hasu_size: $i ).

thf(even_type,type,
    even: $i ).

thf(g__type,type,
    g_: $i ).

thf(t_type,type,
    t: $i ).

thf(r_r__type,type,
    r_r_: $i ).

thf(mod_type,type,
    mod: $i ).

thf(realu_sgn_type,type,
    realu_sgn: $i ).

thf(sum0_type,type,
    sum0: $i > $i > $i ).

thf(inl_type,type,
    inl: $i ).

thf(p_u_c_type,type,
    p_u_c: $i ).

thf(dot_type,type,
    dot: $i ).

thf(el_type,type,
    el: $i ).

thf(vec_type,type,
    vec: $i ).

thf(column_type,type,
    column: $i ).

thf(transp_type,type,
    transp: $i ).

thf(union_type,type,
    union: $i ).

thf(one_type,type,
    one: $i ).

thf(dist_type,type,
    dist: $i ).

thf(a_u_c_type,type,
    a_u_c: $i ).

thf(naddu_ofu_num_type,type,
    naddu_ofu_num: $i ).

thf(basis_type,type,
    basis: $i ).

thf(q_type,type,
    q: $i ).

thf(r0_type,type,
    r0: $i ).

thf(geq_type,type,
    geq: $i ).

thf(ordinal_type,type,
    ordinal: $i ).

thf(gabs_type,type,
    gabs: $i ).

thf(countable_type,type,
    countable: $i ).

thf(support_type,type,
    support: $i ).

thf(c_type,type,
    c: $i ).

thf(d_type,type,
    d: $i ).

thf(disjoint_type,type,
    disjoint: $i ).

thf(bilinear_type,type,
    bilinear: $i ).

thf(lambda_type,type,
    lambda: $i ).

thf(g_type,type,
    g: $i ).

thf(trealu_le_type,type,
    trealu_le: $i ).

thf(hrealu_le_type,type,
    hrealu_le: $i ).

thf(hrealu_add_type,type,
    hrealu_add: $i ).

thf(reverse_type,type,
    reverse: $i ).

thf(append_type,type,
    append: $i ).

thf(inter_type,type,
    inter: $i ).

thf(pre_type,type,
    pre: $i ).

thf(q0_type,type,
    q0: $i ).

thf(all0_type,type,
    all0: $i ).

thf(naddu_add_type,type,
    naddu_add: $i ).

thf(intu_ofu_real_type,type,
    intu_ofu_real: $i ).

thf(realu_ofu_int_type,type,
    realu_ofu_int: $i ).

thf(trealu_neg_type,type,
    trealu_neg: $i ).

thf(intu_gt_type,type,
    intu_gt: $i ).

thf(intu_ge_type,type,
    intu_ge: $i ).

thf(inters_type,type,
    inters: $i ).

thf(numu_divides_type,type,
    numu_divides: $i ).

thf(z_type,type,
    z: $i ).

thf(op_type,type,
    op: $i ).

thf(toset_type,type,
    toset: $i ).

thf(poset_type,type,
    poset: $i ).

thf(l_a_u_c_type,type,
    l_a_u_c: $i ).

thf(fst_type,type,
    fst: $i ).

thf(snd_type,type,
    snd: $i ).

thf(admissible_type,type,
    admissible: $i ).

thf(matrixu_neg_type,type,
    matrixu_neg: $i ).

thf(t_u_c_type,type,
    t_u_c: $i ).

thf(infinite_type,type,
    infinite: $i ).

thf(matrixu_add_type,type,
    matrixu_add: $i ).

thf(cross0_type,type,
    cross0: $i ).

thf(recspace_type,type,
    recspace: $i > $i ).

thf(constr_type,type,
    constr: $i ).

thf(bottom_type,type,
    bottom: $i ).

thf(c0_type,type,
    c0: $i ).

thf(chain_type,type,
    chain: $i ).

thf(trealu_inv_type,type,
    trealu_inv: $i ).

thf(u_type,type,
    u: $i ).

thf(inr_type,type,
    inr: $i ).

thf(a_a__type,type,
    a_a_: $i ).

thf(numu_mod_type,type,
    numu_mod: $i ).

thf(intu_mod_type,type,
    intu_mod: $i ).

thf(from_type,type,
    from: $i ).

thf(zbot_type,type,
    zbot: $i ).

thf(injp_type,type,
    injp: $i ).

thf(injn_type,type,
    injn: $i ).

thf(setu_ofu_list_type,type,
    setu_ofu_list: $i ).

thf(butlast_type,type,
    butlast: $i ).

thf(finite_sum_type,type,
    finite_sum: $i > $i > $i ).

thf(mku_finiteu_sum_type,type,
    mku_finiteu_sum: $i ).

thf(destu_finiteu_sum_type,type,
    destu_finiteu_sum: $i ).

thf(vectoru_neg_type,type,
    vectoru_neg: $i ).

thf(fcons_type,type,
    fcons: $i ).

thf(mem_type,type,
    mem: $i ).

thf(assoc_type,type,
    assoc: $i ).

thf(pastecart_type,type,
    pastecart: $i ).

thf(naddu_le_type,type,
    naddu_le: $i ).

thf(itset_type,type,
    itset: $i ).

thf(y2_type,type,
    y2: $i ).

thf(x2_type,type,
    x2: $i ).

thf(isu_nadd_type,type,
    isu_nadd: $i ).

thf(realu_min_type,type,
    realu_min: $i ).

thf(realu_max_type,type,
    realu_max: $i ).

thf(wf_type,type,
    wf: $i ).

thf(inseg_type,type,
    inseg: $i ).

thf(e0_type,type,
    e0: $i ).

thf(div0_type,type,
    div0: $i ).

thf(rem_type,type,
    rem: $i ).

thf(itlist_type,type,
    itlist: $i ).

thf(ex0_type,type,
    ex0: $i ).

thf(a_type,type,
    a: $i ).

thf(b_type,type,
    b: $i ).

thf(d0_type,type,
    d0: $i ).

thf(linseg_type,type,
    linseg: $i ).

thf(listu_ofu_set_type,type,
    listu_ofu_set: $i ).

thf(numsum_type,type,
    numsum: $i ).

thf(l_u_c_type,type,
    l_u_c: $i ).

thf(vectoru_matrixu_mul_type,type,
    vectoru_matrixu_mul: $i ).

thf(tailadmissible_type,type,
    tailadmissible: $i ).

thf(e__type,type,
    e_: $i ).

thf(pairwise0_type,type,
    pairwise0: $i ).

thf(invertible_type,type,
    invertible: $i ).

thf(matrixu_inv_type,type,
    matrixu_inv: $i ).

thf(measure0_type,type,
    measure0: $i ).

thf(zconstr_type,type,
    zconstr: $i ).

thf(inja_type,type,
    inja: $i ).

thf(injf_type,type,
    injf: $i ).

thf(i1_type,type,
    i1: $i ).

thf(numu_coprime_type,type,
    numu_coprime: $i ).

thf(intu_coprime_type,type,
    intu_coprime: $i ).

thf(orthogonal_type,type,
    orthogonal: $i ).

thf(l_l__type,type,
    l_l_: $i ).

thf(l_l_l__type,type,
    l_l_l_: $i ).

thf(matrixu_sub_type,type,
    matrixu_sub: $i ).

thf(sndcart_type,type,
    sndcart: $i ).

thf(fstcart_type,type,
    fstcart: $i ).

thf(destu_cart_type,type,
    destu_cart: $i ).

thf(hull_type,type,
    hull: $i ).

thf(superadmissible_type,type,
    superadmissible: $i ).

thf(numpair_type,type,
    numpair: $i ).

thf(pairwise_type,type,
    pairwise: $i ).

thf(u_seqpattern_type,type,
    u_seqpattern: $i ).

thf(g_a_u_c_type,type,
    g_a_u_c: $i ).

thf(null_type,type,
    null: $i ).

thf(ind0_type,type,
    ind0: $i ).

thf(mku_num_type,type,
    mku_num: $i ).

thf(indu_0_type,type,
    indu_0: $i ).

thf(columns_type,type,
    columns: $i ).

thf(indu_suc_type,type,
    indu_suc: $i ).

thf(destu_num_type,type,
    destu_num: $i ).

thf(realu_ge_type,type,
    realu_ge: $i ).

thf(u_destu_rec_type,type,
    u_destu_rec: $i ).

thf(u_mku_rec_type,type,
    u_mku_rec: $i ).

thf(i0_type,type,
    i0: $i ).

thf(u_guardedu_pattern_type,type,
    u_guardedu_pattern: $i ).

thf(li_0_type,type,
    li_0: $i ).

thf(integer_type,type,
    integer: $i ).

thf(u_unguardedu_pattern_type,type,
    u_unguardedu_pattern: $i ).

thf(absu_prod_type,type,
    absu_prod: $i ).

thf(mku_pair_type,type,
    mku_pair: $i ).

thf(char_type,type,
    char: $i ).

thf(ascii_type,type,
    ascii: $i ).

thf(zrecspace_type,type,
    zrecspace: $i ).

thf(gi__type,type,
    gi_: $i ).

thf(fi__type,type,
    fi_: $i ).

thf(iso0_type,type,
    iso0: $i ).

thf(minimal_type,type,
    minimal: $i ).

thf(numleft_type,type,
    numleft: $i ).

thf(numright_type,type,
    numright: $i ).

thf(outl_type,type,
    outl: $i ).

thf(let_type,type,
    let: $i ).

thf(inj_type,type,
    inj: $i ).

thf(onto_type,type,
    onto: $i ).

thf(u_function_type,type,
    u_function: $i ).

thf(mku_cart_type,type,
    mku_cart: $i ).

thf(j_type,type,
    j: $i ).

thf(letu_end_type,type,
    letu_end: $i ).

thf(curry_type,type,
    curry: $i ).

thf(repu_prod_type,type,
    repu_prod: $i ).

thf(numfst_type,type,
    numfst: $i ).

thf(numsnd_type,type,
    numsnd: $i ).

thf(adjoint_type,type,
    adjoint: $i ).

thf(u_falsityu__type,type,
    u_falsityu_: $i ).

thf(g_u_c_type,type,
    g_u_c: $i ).

thf(option_type,type,
    option: $i > $i ).

thf(none_type,type,
    none: $i ).

thf(some_type,type,
    some: $i ).

thf(choice0_type,type,
    choice0: $i ).

thf(oneu_one_type,type,
    oneu_one: $i ).

thf(decimal_type,type,
    decimal: $i ).

thf(numu_rep_type,type,
    numu_rep: $i ).

thf(uncurry_type,type,
    uncurry: $i ).

thf(bij_type,type,
    bij: $i ).

thf(surj_type,type,
    surj: $i ).

thf(oneu_abs_type,type,
    oneu_abs: $i ).

thf(oneu_rep_type,type,
    oneu_rep: $i ).

thf(rest_type,type,
    rest: $i ).

thf(realu_mod_type,type,
    realu_mod: $i ).

thf(outr_type,type,
    outr: $i ).

thf(passoc_type,type,
    passoc: $i ).

thf(sing_type,type,
    sing: $i ).

thf(fnil_type,type,
    fnil: $i ).

thf(1835,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_GT) ).

thf(14404,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1835]) ).

thf(2698,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ vectoru_matrixu_mul ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MATRIXu_MULu_TRANSP) ).

thf(19117,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ vectoru_matrixu_mul ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2698]) ).

thf(1768,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( cart @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ ( cart @ A @ B ) @ ( cart @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ A @ B ) @ ( fun @ ( cart @ A @ B ) @ ( cart @ A @ B ) ) ) ) @ cond ) @ ( s @ bool @ b0 ) ) ) @ ( s @ ( cart @ A @ B ) @ x ) ) ) @ ( s @ ( cart @ A @ B ) @ y ) ) ) ) ) @ ( s @ num @ i0 ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ b0 ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ x ) ) ) @ ( s @ num @ i0 ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ y ) ) ) @ ( s @ num @ i0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONDu_COMPONENT) ).

thf(13950,plain,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( cart @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ ( cart @ A @ B ) @ ( cart @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ A @ B ) @ ( fun @ ( cart @ A @ B ) @ ( cart @ A @ B ) ) ) ) @ cond ) @ ( s @ bool @ b0 ) ) ) @ ( s @ ( cart @ A @ B ) @ x ) ) ) @ ( s @ ( cart @ A @ B ) @ y ) ) ) ) ) @ ( s @ num @ i0 ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ b0 ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ x ) ) ) @ ( s @ num @ i0 ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ y ) ) ) @ ( s @ num @ i0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1768]) ).

thf(2197,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFACTu_MONO) ).

thf(16466,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2197]) ).

thf(2525,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
         => ( ( s @ B @ D )
            = ( s @ B @ E ) ) )
    <=> ! [D: $i,E: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
        <=> ( ( s @ B @ D )
            = ( s @ B @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJECTIVEu_ALT) ).

thf(18180,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i,E: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
           => ( ( s @ B @ D )
              = ( s @ B @ E ) ) )
       => ! [D: $i,E: $i] :
            ( ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
             => ( ( s @ B @ D )
                = ( s @ B @ E ) ) )
            & ( ( ( s @ B @ D )
                = ( s @ B @ E ) )
             => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) ) )
      & ( ! [D: $i,E: $i] :
            ( ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
             => ( ( s @ B @ D )
                = ( s @ B @ E ) ) )
            & ( ( ( s @ B @ D )
                = ( s @ B @ E ) )
             => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) )
       => ! [D: $i,E: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
           => ( ( s @ B @ D )
              = ( s @ B @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2525]) ).

thf(363,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_MUL) ).

thf(5385,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).

thf(423,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_RID) ).

thf(5810,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).

thf(2717,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
         => ( ( s @ B @ E )
            = ( s @ B @ F ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_IMAGEu_INJu_EQ) ).

thf(19204,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
         => ( ( s @ B @ E )
            = ( s @ B @ F ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2717]) ).

thf(2080,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ B @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ B @ F ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
            <=> ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ E ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ H ) ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSIMPLEu_IMAGEu_GEN) ).

thf(15767,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ B @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ B @ F ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ B @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
               => ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ E ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ H ) ) ) ) ) ) )
              & ( ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ E ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ H ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2080]) ).

thf(3158,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) )
      = ( s @ ( fun @ A @ bool ) @ univ ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERSu_0) ).

thf(21624,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) )
      = ( s @ ( fun @ A @ bool ) @ univ ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3158]) ).

thf(693,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_ADDL) ).

thf(7304,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[693]) ).

thf(1169,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_MUL) ).

thf(10007,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1169]) ).

thf(3177,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_LEu_ADDu_RCANCEL) ).

thf(21729,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3177]) ).

thf(2906,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GEu_conjunct9) ).

thf(20256,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2906]) ).

thf(2625,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_OVERu_INTER) ).

thf(18768,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2625]) ).

thf(974,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ C ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMu_EL) ).

thf(8960,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ C ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[974]) ).

thf(1959,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ toset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
          & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aOEP) ).

thf(15170,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ toset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
          & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1959]) ).

thf(1072,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_ADD2) ).

thf(9405,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1072]) ).

thf(2404,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
    <=> ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
           => ( ( s @ A @ C )
              = ( s @ A @ D ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
              | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) )
        & ! [C: $i] :
            ( ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
              & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                & ! [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',awoset) ).

thf(17507,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
       => ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
          & ! [C: $i,D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) )
          & ! [C: $i] :
              ( ( ! [D: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
                & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
             => ? [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                  & ! [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ) ) ) )
      & ( ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
          & ! [C: $i,D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) )
          & ! [C: $i] :
              ( ( ! [D: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
                & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
             => ? [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                  & ! [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2404]) ).

thf(1625,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                    = ( s @ A @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ num ) @ G ) ) ) @ ( s @ A @ H ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) )
             => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_GROUP) ).

thf(13024,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ num ) @ G ) ) ) @ ( s @ A @ H ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) )
             => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1625]) ).

thf(1916,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
    ? [F: $i] :
    ! [G: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ D ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_RECu_TAIL) ).

thf(14885,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
    ? [F: $i] :
    ! [G: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ D ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1916]) ).

thf(2424,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTEu_TRANS) ).

thf(17639,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2424]) ).

thf(369,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ bool ) ) @ bool ) ) ) @ a_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ numu_mod ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) ) ) @ a_a_ ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ int @ bool ) ) ) @ intu_mod ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_congruent) ).

thf(5469,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ bool ) ) @ bool ) ) ) @ a_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ numu_mod ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) ) ) @ a_a_ ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ int @ bool ) ) ) @ intu_mod ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).

thf(1272,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_1) ).

thf(10647,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1272]) ).

thf(2798,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPSUBSETu_MEMBER) ).

thf(19672,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2798]) ).

thf(2986,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
       => ( p @ ( s @ bool @ C ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_IMPu_EXISTSu_THM) ).

thf(20690,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ C ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ C ) ) )
       => ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2986]) ).

thf(2449,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) )
    <=> ( ( ( s @ A @ E )
          = ( s @ A @ C ) )
        & ( ( s @ B @ F )
          = ( s @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amku_pairu_def) ).

thf(17803,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) )
       => ( ( ( s @ A @ E )
            = ( s @ A @ C ) )
          & ( ( s @ B @ F )
            = ( s @ B @ D ) ) ) )
      & ( ( ( ( s @ A @ E )
            = ( s @ A @ C ) )
          & ( ( s @ B @ F )
            = ( s @ B @ D ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2449]) ).

thf(321,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) )
        & ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_LT) ).

thf(5116,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) )
        & ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).

thf(1638,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
            & ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
            & ! [F: $i,G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
                  & ( ( s @ ( fun @ A @ bool ) @ F )
                   != ( s @ ( fun @ A @ bool ) @ G ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
               => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ H ) ) )
                  = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_UNIONSu_NONZERO) ).

thf(13162,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
            & ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
            & ! [F: $i,G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
                  & ( ( s @ ( fun @ A @ bool ) @ F )
                   != ( s @ ( fun @ A @ bool ) @ G ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
               => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ H ) ) )
                  = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1638]) ).

thf(691,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_LE) ).

thf(7272,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[691]) ).

thf(1643,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SUBSETu_SIMPLE) ).

thf(13199,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1643]) ).

thf(3066,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_CAUCHYu_SCHWARZu_ABS) ).

thf(21175,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3066]) ).

thf(317,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
    <=> ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ) )
        | ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_EQu_1) ).

thf(5088,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ) )
          | ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ) )
          | ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(1550,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_COMBINEu_L) ).

thf(12597,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1550]) ).

thf(1576,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_UNIONu_NONZERO) ).

thf(12710,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1576]) ).

thf(1539,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_ADD) ).

thf(12490,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1539]) ).

thf(788,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ vectoru_matrixu_mul ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',avectoru_matrixu_mul) ).

thf(7890,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ vectoru_matrixu_mul ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[788]) ).

thf(1977,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_UNIV) ).

thf(15278,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1977]) ).

thf(163,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_RINV) ).

thf(4005,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(1296,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ C )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_ADDu_MOD) ).

thf(10785,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ C )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1296]) ).

thf(325,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i,I: $i,J: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) ) @ E ) @ ( s @ ( fun @ D @ num ) @ F ) ) ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ F ) @ ( s @ D @ J ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ G ) @ ( s @ D @ J ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ D @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i,J: $i] :
              ( ( s @ ( fun @ D @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ D @ num ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
              = ( s @ ( fun @ D @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) ) @ E ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ D @ num ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) )
         => ! [G: $i] :
              ( ! [H: $i,I: $i,J: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ num ) @ J ) ) ) )
                <=> ! [K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ geq ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ num ) @ J ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ D @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ L ) ) ) @ ( s @ num @ K ) ) ) ) ) ) )
             => ! [H: $i] :
                  ( ! [I: $i,J: $i] :
                      ( ( s @ ( fun @ ( prod @ num @ D ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
                      = ( s @ ( fun @ ( prod @ num @ D ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) ) ) )
                 => ! [I: $i] :
                      ( ! [J: $i,K: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) ) @ I ) @ ( s @ ( fun @ D @ A ) @ J ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) ) ) )
                        <=> ! [L: $i,M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ J ) @ ( s @ D @ M ) ) ) ) ) ) )
                     => ! [J: $i] :
                          ( ! [K: $i,L: $i,M: $i,N: $i,O: $i] :
                              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ D @ num ) @ K ) ) ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) ) ) )
                            <=> ! [P: $i,Q: $i] :
                                ? [R: $i] :
                                  ( ( ( p @ ( s @ bool @ R ) )
                                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ K ) @ ( s @ D @ Q ) ) ) ) ) @ ( s @ num @ P ) ) ) )
                                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ P ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ L ) @ ( s @ D @ Q ) ) ) ) ) )
                                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ Q ) ) ) ) ) )
                                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ P ) ) ) @ ( s @ D @ Q ) ) ) ) ) ) ) @ ( s @ bool @ R ) ) ) ) ) )
                         => ! [K: $i] :
                              ( ! [L: $i,M: $i,N: $i,O: $i] :
                                  ( ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ D @ num ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) )
                                  = ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ D @ num ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) ) ) ) )
                             => ! [L: $i,M: $i,N: $i,O: $i,P: $i,Q: $i] :
                                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ D @ num ) @ P ) ) ) @ ( s @ ( fun @ D @ num ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) ) @ I ) @ ( s @ ( fun @ D @ A ) @ N ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ O ) ) ) ) ) )
                                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ D @ A ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ num ) @ P ) ) ) @ ( s @ ( fun @ D @ num ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ O ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_NSUM) ).

thf(5136,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i,I: $i,J: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) ) @ E ) @ ( s @ ( fun @ D @ num ) @ F ) ) ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ F ) @ ( s @ D @ J ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ G ) @ ( s @ D @ J ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ D @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i,J: $i] :
              ( ( s @ ( fun @ D @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ D @ num ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
              = ( s @ ( fun @ D @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) ) @ E ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ D @ num ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) )
         => ! [G: $i] :
              ( ! [H: $i,I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ num ) @ J ) ) ) )
                   => ! [K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ geq ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ num ) @ J ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ D @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ L ) ) ) @ ( s @ num @ K ) ) ) ) ) ) )
                  & ( ! [K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ geq ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ num ) @ J ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ D @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ L ) ) ) @ ( s @ num @ K ) ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ num ) @ J ) ) ) ) ) )
             => ! [H: $i] :
                  ( ! [I: $i,J: $i] :
                      ( ( s @ ( fun @ ( prod @ num @ D ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
                      = ( s @ ( fun @ ( prod @ num @ D ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ num ) @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) ) ) )
                 => ! [I: $i] :
                      ( ! [J: $i,K: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) ) @ I ) @ ( s @ ( fun @ D @ A ) @ J ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) ) ) )
                           => ! [L: $i,M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ J ) @ ( s @ D @ M ) ) ) ) ) ) )
                          & ( ! [L: $i,M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ J ) @ ( s @ D @ M ) ) ) ) ) )
                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) ) @ I ) @ ( s @ ( fun @ D @ A ) @ J ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) ) ) ) ) )
                     => ! [J: $i] :
                          ( ! [K: $i,L: $i,M: $i,N: $i,O: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ D @ num ) @ K ) ) ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) ) ) )
                               => ! [P: $i,Q: $i] :
                                  ? [R: $i] :
                                    ( ( ( p @ ( s @ bool @ R ) )
                                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ K ) @ ( s @ D @ Q ) ) ) ) ) @ ( s @ num @ P ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ P ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ L ) @ ( s @ D @ Q ) ) ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ Q ) ) ) ) ) )
                                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ K ) @ ( s @ D @ Q ) ) ) ) ) @ ( s @ num @ P ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ P ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ L ) @ ( s @ D @ Q ) ) ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ Q ) ) ) ) )
                                     => ( p @ ( s @ bool @ R ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ P ) ) ) @ ( s @ D @ Q ) ) ) ) ) ) ) @ ( s @ bool @ R ) ) ) ) ) )
                              & ( ! [P: $i,Q: $i] :
                                  ? [R: $i] :
                                    ( ( ( p @ ( s @ bool @ R ) )
                                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ K ) @ ( s @ D @ Q ) ) ) ) ) @ ( s @ num @ P ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ P ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ L ) @ ( s @ D @ Q ) ) ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ Q ) ) ) ) ) )
                                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ K ) @ ( s @ D @ Q ) ) ) ) ) @ ( s @ num @ P ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ P ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ L ) @ ( s @ D @ Q ) ) ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ Q ) ) ) ) )
                                     => ( p @ ( s @ bool @ R ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ P ) ) ) @ ( s @ D @ Q ) ) ) ) ) ) ) @ ( s @ bool @ R ) ) ) ) )
                               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ D @ num ) @ K ) ) ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) ) ) ) ) )
                         => ! [K: $i] :
                              ( ! [L: $i,M: $i,N: $i,O: $i] :
                                  ( ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ D @ num ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) )
                                  = ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ D @ num ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) ) ) ) )
                             => ! [L: $i,M: $i,N: $i,O: $i,P: $i,Q: $i] :
                                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ D @ num ) @ P ) ) ) @ ( s @ ( fun @ D @ num ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) ) @ I ) @ ( s @ ( fun @ D @ A ) @ N ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ num ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ O ) ) ) ) ) )
                                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ D @ A ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ num ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ num ) @ P ) ) ) @ ( s @ ( fun @ D @ num ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ num ) ) ) @ O ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).

thf(832,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
        <=> ? [G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
          = ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amulu_c) ).

thf(8154,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
           => ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
          & ( ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
          = ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[832]) ).

thf(2068,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_DELETEu_IMP) ).

thf(15720,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2068]) ).

thf(3173,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        & ( p @ ( s @ bool @ C ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_ANDu_FORALLu_THM) ).

thf(21710,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ C ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ C ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ C ) ) )
       => ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3173]) ).

thf(738,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
        <=> ! [F: $i,G: $i] :
            ? [H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alinseg) ).

thf(7569,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
           => ! [F: $i,G: $i] :
              ? [H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
          & ( ! [F: $i,G: $i] :
              ? [H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[738]) ).

thf(1998,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ! [D: $i,E: $i] :
            ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ D )
              = ( s @ ( cart @ real @ B ) @ E ) ) )
      <=> ! [D: $i] :
            ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ D )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_INJECTIVEu_0) ).

thf(15418,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ( ! [D: $i,E: $i] :
              ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
                = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
             => ( ( s @ ( cart @ real @ B ) @ D )
                = ( s @ ( cart @ real @ B ) @ E ) ) )
         => ! [D: $i] :
              ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
                = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
             => ( ( s @ ( cart @ real @ B ) @ D )
                = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
        & ( ! [D: $i] :
              ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
                = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
             => ( ( s @ ( cart @ real @ B ) @ D )
                = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
         => ! [D: $i,E: $i] :
              ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
                = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
             => ( ( s @ ( cart @ real @ B ) @ D )
                = ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1998]) ).

thf(190,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_DIFFu_GEN) ).

thf(4262,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(632,axiom,
    ( ( ( ( p @ ( s @ bool @ a ) )
       => ( p @ ( s @ bool @ b ) ) )
      & ( ( p @ ( s @ bool @ c0 ) )
       => ( p @ ( s @ bool @ d0 ) ) ) )
   => ( ( ( p @ ( s @ bool @ a ) )
        | ( p @ ( s @ bool @ c0 ) ) )
     => ( ( p @ ( s @ bool @ b ) )
        | ( p @ ( s @ bool @ d0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOu_OR) ).

thf(6976,plain,
    ( ( ( ( p @ ( s @ bool @ a ) )
       => ( p @ ( s @ bool @ b ) ) )
      & ( ( p @ ( s @ bool @ c0 ) )
       => ( p @ ( s @ bool @ d0 ) ) ) )
   => ( ( ( p @ ( s @ bool @ a ) )
        | ( p @ ( s @ bool @ c0 ) ) )
     => ( ( p @ ( s @ bool @ b ) )
        | ( p @ ( s @ bool @ d0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[632]) ).

thf(2861,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ B ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NOTu_EQ) ).

thf(20014,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ A )
         != ( s @ real @ B ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) )
       => ( ( s @ real @ A )
         != ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2861]) ).

thf(37,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) )
        = ( s @ real @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_LINVu_UNIQ) ).

thf(3399,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) )
        = ( s @ real @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(2583,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ B ) @ bool ) @ ( fun @ B @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ B @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ B @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ B @ B ) @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ ( prod @ B @ B ) @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ B @ x ) ) ) )
    <=> ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ B ) @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ ( prod @ B @ B ) @ bool ) @ E ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ B ) @ bool ) @ ( fun @ B @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ B @ B ) @ bool ) @ E ) ) ) @ ( s @ B @ x ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_FL) ).

thf(18504,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ A @ x ) ) ) )
       => ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ x ) ) ) ) ) )
      & ( ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ x ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ A @ x ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2583]) ).

thf(3016,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ A )
        = ( s @ num @ B ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSsrnato_eqnu_leq) ).

thf(20850,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3016]) ).

thf(564,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',avectoru_mul) ).

thf(6544,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).

thf(185,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_DIV) ).

thf(4169,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(583,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_BETWEEN) ).

thf(6712,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).

thf(1723,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) )
    <=> ( ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_SUC) ).

thf(13643,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) )
       => ( ( ( s @ ( fun @ A @ bool ) @ B )
           != ( s @ ( fun @ A @ bool ) @ empty ) )
          & ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
      & ( ( ( ( s @ ( fun @ A @ bool ) @ B )
           != ( s @ ( fun @ A @ bool ) @ empty ) )
          & ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1723]) ).

thf(3116,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LTu_FINITEu_INFINITE) ).

thf(21439,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3116]) ).

thf(89,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_OFFSET) ).

thf(3666,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(1016,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIMINDEXu_HASu_SIZEu_FINITEu_SUM) ).

thf(9180,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1016]) ).

thf(2005,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
      = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) ) @ vectoru_matrixu_mul ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_TRANSP) ).

thf(15472,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
      = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) ) @ vectoru_matrixu_mul ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2005]) ).

thf(2782,axiom,
    p @ ( s @ bool @ t ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_UNIQUEu_DEF) ).

thf(19594,plain,
    p @ ( s @ bool @ t ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2782]) ).

thf(2571,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_ADDu_FINITE) ).

thf(18442,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2571]) ).

thf(3026,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_SINGu_NUMSEG) ).

thf(20902,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3026]) ).

thf(2791,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ o ) @ ( s @ ( fun @ B @ B ) @ i1 ) ) ) @ ( s @ ( fun @ A @ B ) @ C ) ) )
        = ( s @ ( fun @ A @ B ) @ C ) )
      & ( ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ B ) ) ) @ o ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ A ) @ i1 ) ) )
        = ( s @ ( fun @ A @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIu_Ou_ID) ).

thf(19623,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ o ) @ ( s @ ( fun @ B @ B ) @ i1 ) ) ) @ ( s @ ( fun @ A @ B ) @ C ) ) )
        = ( s @ ( fun @ A @ B ) @ C ) )
      & ( ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ B ) ) ) @ o ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ A ) @ i1 ) ) )
        = ( s @ ( fun @ A @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2791]) ).

thf(161,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EXPu_conjunct1) ).

thf(3979,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(898,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBOUNDSu_LINEAR) ).

thf(8507,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
       => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).

thf(2940,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLETu_TRANS) ).

thf(20464,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2940]) ).

thf(2268,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ! [D: $i,E: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) )
        & ! [D: $i,E: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADJOINTu_CLAUSES) ).

thf(16808,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ! [D: $i,E: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) )
        & ! [D: $i,E: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2268]) ).

thf(2961,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ D ) ) )
            = ( s @ A @ D ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ A ) @ D ) ) ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
         => ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ A @ A ) @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) )
            = ( s @ ( list @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_EQu_DEGEN) ).

thf(20559,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ A @ D ) ) )
          & ( ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ A @ D ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ A ) @ D ) ) ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
         => ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ A @ A ) @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) )
            = ( s @ ( list @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2961]) ).

thf(1394,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_LMULT) ).

thf(11581,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1394]) ).

thf(3120,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) ) ) )
           => ( ( s @ A @ E )
              = ( s @ A @ F ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ real ) ) ) @ o ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_INJECTION) ).

thf(21450,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) ) ) )
           => ( ( s @ A @ E )
              = ( s @ A @ F ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ real ) ) ) @ o ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3120]) ).

thf(1803,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LE2u_ODD) ).

thf(14220,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1803]) ).

thf(133,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_COMBINEu_R) ).

thf(3863,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(1530,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
    <=> ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ int @ num ) @ numu_ofu_int ) @ ( s @ int @ A ) ) ) ) )
        = ( s @ int @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMu_OFu_INT) ).

thf(12440,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ int @ num ) @ numu_ofu_int ) @ ( s @ int @ A ) ) ) ) )
          = ( s @ int @ A ) ) )
      & ( ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ int @ num ) @ numu_ofu_int ) @ ( s @ int @ A ) ) ) ) )
          = ( s @ int @ A ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1530]) ).

thf(131,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) )
                  = ( s @ A @ F ) ) ) )
      <=> ! [F: $i,G: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
           => ( ( s @ B @ F )
              = ( s @ B @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJECTIVEu_IFFu_INJECTIVEu_GEN) ).

thf(3830,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
             => ? [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) )
                    = ( s @ A @ F ) ) ) )
         => ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ B @ F )
                = ( s @ B @ G ) ) ) )
        & ( ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ B @ F )
                = ( s @ B @ G ) ) )
         => ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
             => ? [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) )
                    = ( s @ A @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(2985,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ A ) ) )
        = ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ B ) ) ) )
    <=> ( ( s @ ind0 @ A )
        = ( s @ ind0 @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINDu_SUCu_INJ) ).

thf(20683,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ A ) ) )
          = ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ B ) ) ) )
       => ( ( s @ ind0 @ A )
          = ( s @ ind0 @ B ) ) )
      & ( ( ( s @ ind0 @ A )
          = ( s @ ind0 @ B ) )
       => ( ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ A ) ) )
          = ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2985]) ).

thf(1793,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ q0 ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) @ ( s @ ( list @ B ) @ li_0 ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ q0 ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) @ ( s @ ( list @ B ) @ li_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOu_ALL2) ).

thf(14139,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ q0 ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) @ ( s @ ( list @ B ) @ li_0 ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ q0 ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) @ ( s @ ( list @ B ) @ li_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1793]) ).

thf(2350,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_INSERT) ).

thf(17175,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2350]) ).

thf(2490,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
     => ? [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTARSKIu_SET) ).

thf(17998,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
     => ? [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2490]) ).

thf(853,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ! [D: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        <=> ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ D ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) )
             => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POLYFUNu_EQu_0) ).

thf(8266,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( ! [D: $i] :
                ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
           => ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ D ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) )
               => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ D ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) )
               => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
           => ! [D: $i] :
                ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[853]) ).

thf(49,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      <=> ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_EQu_0) ).

thf(3449,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
         => ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
        & ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(1183,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ p1 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_ACu_conjunct2) ).

thf(10081,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ p1 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1183]) ).

thf(1137,axiom,
    ! [A: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
        = ( s @ num @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_0) ).

thf(9864,plain,
    ! [A: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
        = ( s @ num @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1137]) ).

thf(634,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ n3 @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ n3 @ bool ) @ univ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIMINDEXu_3) ).

thf(6986,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ n3 @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ n3 @ bool ) @ univ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[634]) ).

thf(914,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_UNION) ).

thf(8646,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[914]) ).

thf(2530,axiom,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ? [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_MAX) ).

thf(18228,plain,
    ! [A: $i] :
      ( ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
          & ? [B: $i] :
            ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ? [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
            & ! [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ C ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      & ( ? [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
            & ! [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ C ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
          & ? [B: $i] :
            ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2530]) ).

thf(1226,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ! [E: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ real @ E ) ) ) ) )
              = ( s @ real @ D ) )
        <=> ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
              = ( s @ real @ D ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) )
               => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POLYFUNu_EQu_CONST) ).

thf(10363,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( ! [E: $i] :
                ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ real @ E ) ) ) ) )
                = ( s @ real @ D ) )
           => ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                = ( s @ real @ D ) )
              & ! [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          & ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                = ( s @ real @ D ) )
              & ! [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
           => ! [E: $i] :
                ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ real @ E ) ) ) ) )
                = ( s @ real @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1226]) ).

thf(472,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ m ) ) ) @ ( s @ int @ n ) ) ) ) ) @ ( s @ int @ p1 ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ m ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ n ) ) ) @ ( s @ int @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_ACu_conjunct1) ).

thf(6042,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ m ) ) ) @ ( s @ int @ n ) ) ) ) ) @ ( s @ int @ p1 ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ m ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ n ) ) ) @ ( s @ int @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).

thf(2137,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_EQ) ).

thf(16124,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2137]) ).

thf(414,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNION) ).

thf(5733,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).

thf(1357,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_HASu_SIZE) ).

thf(11318,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1357]) ).

thf(891,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_sgn) ).

thf(8475,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[891]) ).

thf(2815,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ! [D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_INSERT) ).

thf(19745,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ! [D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2815]) ).

thf(2520,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_ADDu_WELLDEF) ).

thf(18154,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2520]) ).

thf(2787,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_SUBSET) ).

thf(19611,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2787]) ).

thf(2993,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( s @ A @ C )
            = ( s @ A @ D ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_UNWINDu_THM1) ).

thf(20726,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( ( s @ A @ C )
              = ( s @ A @ D ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ! [D: $i] :
            ( ( ( s @ A @ C )
              = ( s @ A @ D ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2993]) ).

thf(1542,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_0) ).

thf(12507,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1542]) ).

thf(2542,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADJOINTu_WORKS) ).

thf(18311,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2542]) ).

thf(1919,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) @ ( s @ real @ C ) ) ) )
        <=> ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) )
            & ! [D: $i] :
                ( ! [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ E ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ D ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ h_ ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asup) ).

thf(14944,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) @ ( s @ real @ C ) ) ) )
           => ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) )
              & ! [D: $i] :
                  ( ! [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ E ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) ) )
          & ( ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) )
              & ! [D: $i] :
                  ( ! [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ E ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ h_ ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1919]) ).

thf(822,axiom,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
      <=> ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ f ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_CLAUSESu_NUMSEGu_conjunct0) ).

thf(8077,plain,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
       => ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ B ) ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ f ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[822]) ).

thf(1399,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ C ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_ADDNEG) ).

thf(11659,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ C ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1399]) ).

thf(2473,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
            & ! [D: $i,E: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ! [D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
             => ( ( s @ ( fun @ A @ bool ) @ C )
                = ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZLu_SUBSETSu_UNIONS) ).

thf(17920,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
            & ! [D: $i,E: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ! [D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
             => ( ( s @ ( fun @ A @ bool ) @ C )
                = ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2473]) ).

thf(1992,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_DIFFSQ) ).

thf(15380,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1992]) ).

thf(2711,axiom,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTEu_ANTISYM) ).

thf(19187,plain,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2711]) ).

thf(1421,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ E ) @ ( s @ D @ F ) ) )
          = ( s @ bool @ t ) )
     => ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ nil ) ) ) @ ( s @ B @ f ) ) ) @ ( s @ C @ x ) ) )
        = ( s @ D @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ D ) @ h_ ) @ ( s @ ( fun @ D @ bool ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCASEWISEu_conjunct0) ).

thf(11790,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ E ) @ ( s @ D @ F ) ) )
          = ( s @ bool @ t ) )
     => ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ nil ) ) ) @ ( s @ B @ f ) ) ) @ ( s @ C @ x ) ) )
        = ( s @ D @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ D ) @ h_ ) @ ( s @ ( fun @ D @ bool ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1421]) ).

thf(2299,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_EMPTY) ).

thf(16950,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2299]) ).

thf(992,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
         => ! [C: $i,D: $i,E: $i,F: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_PARTIALu_PRE) ).

thf(9032,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
         => ! [C: $i,D: $i,E: $i,F: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[992]) ).

thf(2675,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_LIST) ).

thf(18985,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2675]) ).

thf(2679,axiom,
    ! [A: $i,B: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_PSUBSETu_EMPTY) ).

thf(18995,plain,
    ! [A: $i,B: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2679]) ).

thf(1520,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ univ )
      = ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( finite_image @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( finite_image @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( finite_image @ A ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_IMAGEu_IMAGE) ).

thf(12371,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ univ )
      = ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( finite_image @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( finite_image @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( finite_image @ A ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1520]) ).

thf(2772,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAXu_SYM) ).

thf(19533,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2772]) ).

thf(2718,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) )
        <=> ? [I: $i,J: $i,K: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) ) )
     => ! [F: $i,G: $i] :
          ( ? [H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ H ) ) ) ) )
        <=> ? [H: $i,I: $i,J: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_INu_GSPECu_conjunct2) ).

thf(19213,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) )
           => ? [I: $i,J: $i,K: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) ) )
          & ( ? [I: $i,J: $i,K: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) ) ) )
     => ! [F: $i,G: $i] :
          ( ( ? [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ H ) ) ) ) )
           => ? [H: $i,I: $i,J: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) ) ) )
          & ( ? [H: $i,I: $i,J: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) ) )
           => ? [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ H ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2718]) ).

thf(2019,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPPORTu_SUPPORT) ).

thf(15532,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2019]) ).

thf(2801,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_ADDL) ).

thf(19687,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2801]) ).

thf(65,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
        & ! [C: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ ( fun @ A @ bool ) @ C )
               != ( s @ ( fun @ A @ bool ) @ empty ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_INDUCTu_DELETE) ).

thf(3534,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
        & ! [C: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ ( fun @ A @ bool ) @ C )
               != ( s @ ( fun @ A @ bool ) @ empty ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(1056,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ m ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_ACu_conjunct0) ).

thf(9342,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1056]) ).

thf(1818,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) ) )
     => ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ C ) )
      <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_EQ) ).

thf(14296,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) ) )
     => ( ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
            = ( s @ ( fun @ A @ bool ) @ C ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
            = ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1818]) ).

thf(1927,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_SUB) ).

thf(15019,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1927]) ).

thf(959,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_0) ).

thf(8855,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[959]) ).

thf(1560,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
        ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
          <=> ( ( s @ A @ E )
              = ( s @ A @ D ) ) )
          & ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ ( cart @ real @ B ) @ b0 ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ A @ E ) ) ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ b0 ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_DELTA) ).

thf(12634,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
        ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
           => ( ( s @ A @ E )
              = ( s @ A @ D ) ) )
          & ( ( ( s @ A @ E )
              = ( s @ A @ D ) )
           => ( p @ ( s @ bool @ F ) ) )
          & ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ ( cart @ real @ B ) @ b0 ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ A @ E ) ) ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ b0 ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1560]) ).

thf(475,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_SUB2) ).

thf(6063,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).

thf(2111,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_CLAUSESu_conjunct7) ).

thf(15980,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2111]) ).

thf(568,axiom,
    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) )
  <=> ! [A: $i,B: $i] :
        ( ( ( ( ( s @ num @ n )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( ( s @ num @ B )
              = ( s @ num @ m ) ) )
          | ( ( ( s @ num @ m )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ n ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVMODu_ELIMu_THM) ).

thf(6576,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) )
     => ! [A: $i,B: $i] :
          ( ( ( ( ( s @ num @ n )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ B )
                = ( s @ num @ m ) ) )
            | ( ( ( s @ num @ m )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ n ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    & ( ! [A: $i,B: $i] :
          ( ( ( ( ( s @ num @ n )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ B )
                = ( s @ num @ m ) ) )
            | ( ( ( s @ num @ m )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ n ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).

thf(1218,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_DELETE) ).

thf(10299,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1218]) ).

thf(1100,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ B @ b0 ) ) )
      = ( s @ B @ b0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITLISTu_conjunct0) ).

thf(9573,plain,
    ! [A: $i,B: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ B @ b0 ) ) )
      = ( s @ B @ b0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1100]) ).

thf(1749,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
          <=> ( ( s @ A @ D )
              = ( s @ A @ C ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ real ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ real @ b0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ real ) ) @ B ) @ ( s @ A @ D ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ real @ b0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_DELTA) ).

thf(13834,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
           => ( ( s @ A @ D )
              = ( s @ A @ C ) ) )
          & ( ( ( s @ A @ D )
              = ( s @ A @ C ) )
           => ( p @ ( s @ bool @ E ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ real ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ real @ b0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ real ) ) @ B ) @ ( s @ A @ D ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ real @ b0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1749]) ).

thf(2495,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
        = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_LZERO) ).

thf(18020,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
        = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2495]) ).

thf(2378,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_INDUCT) ).

thf(17310,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2378]) ).

thf(1351,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ inter ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFROMu_INTERu_NUMSEGu_GEN) ).

thf(11256,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ inter ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1351]) ).

thf(1921,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
                = ( s @ A @ G ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) )
                = ( s @ B @ G ) ) ) ) )
     => ! [G: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIJECTIONSu_HASu_SIZEu_EQ) ).

thf(14976,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
                = ( s @ A @ G ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) )
                = ( s @ B @ G ) ) ) ) )
     => ! [G: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1921]) ).

thf(1490,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_LID) ).

thf(12129,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1490]) ).

thf(2691,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_INC) ).

thf(19088,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2691]) ).

thf(1563,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ G ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_IMAGEu_INJu_EQ) ).

thf(12649,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ G ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1563]) ).

thf(619,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) ) ) )
        <=> ! [G: $i,H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) ) ) ) ) )
        <=> ? [F: $i,G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_TRIPLEDu_THM) ).

thf(6918,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) ) ) )
           => ! [G: $i,H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) )
          & ( ! [G: $i,H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) ) ) ) ) )
     => ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) ) ) ) ) )
           => ? [F: $i,G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) )
          & ( ? [F: $i,G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).

thf(879,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_INVu_EQ) ).

thf(8434,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[879]) ).

thf(205,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_EQu_0) ).

thf(4339,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(1622,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ nil ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALL2u_conjunct0) ).

thf(13007,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ nil ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1622]) ).

thf(1737,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_LZERO) ).

thf(13712,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1737]) ).

thf(3055,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
         => ( ( s @ B @ D )
            = ( s @ B @ E ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFINITEu_IMAGEu_INJ) ).

thf(21114,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
         => ( ( s @ B @ D )
            = ( s @ B @ E ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3055]) ).

thf(1876,axiom,
    ( ( ( p @ ( s @ bool @ p1 ) )
      | ( p @ ( s @ bool @ q ) ) )
  <=> ( ( p @ ( s @ bool @ q ) )
      | ( p @ ( s @ bool @ p1 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJu_ACIu_conjunct0) ).

thf(14625,plain,
    ( ( ( ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ q ) ) )
     => ( ( p @ ( s @ bool @ q ) )
        | ( p @ ( s @ bool @ p1 ) ) ) )
    & ( ( ( p @ ( s @ bool @ q ) )
        | ( p @ ( s @ bool @ p1 ) ) )
     => ( ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ q ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1876]) ).

thf(550,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
           => ( ( s @ B @ E )
              = ( s @ B @ F ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_IMAGEu_INJ) ).

thf(6495,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
           => ( ( s @ B @ E )
              = ( s @ B @ F ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).

thf(640,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      = ( s @ num @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_RADDu_0) ).

thf(7008,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      = ( s @ num @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[640]) ).

thf(3028,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_MUL2u_ABSORBu_LE) ).

thf(20922,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3028]) ).

thf(142,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SGNu_EQu_conjunct2) ).

thf(3905,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(941,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_UNIONu_LE) ).

thf(8797,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[941]) ).

thf(3009,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ADDu_SYM) ).

thf(20813,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3009]) ).

thf(1973,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ gabs ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aGABSu_DEF) ).

thf(15267,plain,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ gabs ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1973]) ).

thf(746,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) )
        <=> ? [G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ( ! [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                  & ! [I: $i] :
                      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                     => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                        = ( s @ A @ d ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ A ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_FUNSPACE) ).

thf(7641,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) )
           => ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                    & ! [I: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                          = ( s @ A @ d ) ) ) ) )
                & ( ( ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                    & ! [I: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                          = ( s @ A @ d ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) ) ) )
          & ( ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                    & ! [I: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                          = ( s @ A @ d ) ) ) ) )
                & ( ( ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                    & ! [I: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                          = ( s @ A @ d ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ A ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[746]) ).

thf(1480,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
      <=> ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ div0 ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
            = ( s @ int @ A ) ) ) )
      & ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
          <=> ( ( s @ int @ B )
              = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
          & ? [E: $i] :
              ( ( ( p @ ( s @ bool @ E ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) )
                  & ( ( s @ int @ A )
                    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ div0 ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ C ) ) ) @ ( s @ bool @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_DIVISIONu_0) ).

thf(12084,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
       => ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ div0 ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
            = ( s @ int @ A ) ) ) )
      & ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ div0 ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
            = ( s @ int @ A ) ) )
       => ( p @ ( s @ bool @ C ) ) )
      & ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
           => ( ( s @ int @ B )
              = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
          & ( ( ( s @ int @ B )
              = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
           => ( p @ ( s @ bool @ D ) ) )
          & ? [E: $i] :
              ( ( ( p @ ( s @ bool @ E ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) )
                  & ( ( s @ int @ A )
                    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ div0 ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) )
              & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) )
                  & ( ( s @ int @ A )
                    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ div0 ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ E ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ C ) ) ) @ ( s @ bool @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1480]) ).

thf(2369,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LTu_IMPu_LE) ).

thf(17278,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2369]) ).

thf(2225,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ empty )
     != ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_EMPTYu_INSERT) ).

thf(16619,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ empty )
     != ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2225]) ).

thf(301,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LEu_1) ).

thf(5000,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(2876,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTEu_TRANS) ).

thf(20136,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2876]) ).

thf(3109,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIFFu_UNIV) ).

thf(21418,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3109]) ).

thf(659,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
        & ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_COUNTABLEu_ASu_IMAGE) ).

thf(7132,plain,
    ! [A: $i] :
      ( ( ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
          & ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
          & ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) ) )
       => ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[659]) ).

thf(2910,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) )
    <=> ( ( s @ num @ C )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJN) ).

thf(20268,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) )
       => ( ( s @ num @ C )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ C )
          = ( s @ num @ B ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2910]) ).

thf(2216,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ? [D: $i] :
        ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_BOUNDED) ).

thf(16566,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ? [D: $i] :
        ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2216]) ).

thf(24,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( list @ A ) @ B )
       != ( s @ ( list @ A ) @ nil ) )
     => ( ( s @ ( list @ A ) @ B )
        = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONSu_HDu_TL) ).

thf(3285,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( list @ A ) @ B )
       != ( s @ ( list @ A ) @ nil ) )
     => ( ( s @ ( list @ A ) @ B )
        = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(2425,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_ODD) ).

thf(17641,plain,
    ! [A: $i] :
      ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2425]) ).

thf(2931,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ D ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
          = ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
    <=> ( ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ D )
        = ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIREDu_EXT) ).

thf(20400,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i,G: $i] :
            ( ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ D ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
            = ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
       => ( ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ D )
          = ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) )
      & ( ( ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ D )
          = ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) )
       => ! [F: $i,G: $i] :
            ( ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ D ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
            = ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2931]) ).

thf(1052,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_RMUL) ).

thf(9324,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1052]) ).

thf(2586,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_ELIMu_THMu_conjunct3) ).

thf(18549,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) ) )
          & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2586]) ).

thf(52,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_PSUBSET) ).

thf(3460,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(1637,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_CLAUSESu_RIGHT) ).

thf(13159,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1637]) ).

thf(612,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) @ ( s @ B @ b0 ) ) )
      = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) @ ( s @ A @ h0 ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) @ ( s @ B @ b0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITLISTu_conjunct1) ).

thf(6900,plain,
    ! [A: $i,B: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) @ ( s @ B @ b0 ) ) )
      = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) @ ( s @ A @ h0 ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) @ ( s @ B @ b0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[612]) ).

thf(888,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_mulu_th) ).

thf(8463,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[888]) ).

thf(2762,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) )
        <=> ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPu_EQ) ).

thf(19438,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) )
           => ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) )
          & ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) )
           => ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2762]) ).

thf(143,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_DIVu_SQRT) ).

thf(3912,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(3054,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_lt) ).

thf(21111,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3054]) ).

thf(1279,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_EQu_0) ).

thf(10705,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1279]) ).

thf(1690,axiom,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) )
           => ( ( s @ A @ C )
              = ( s @ A @ D ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ C ) ) ) )
         => ? [E: $i] :
              ( ( ( s @ ( fun @ A @ bool ) @ D )
                = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
              & ! [F: $i,G: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ F ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ G ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) )
                 => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ E ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ E ) @ ( s @ num @ F ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTOPOLOGICALu_SORT) ).

thf(13462,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) )
           => ( ( s @ A @ C )
              = ( s @ A @ D ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ C ) ) ) )
         => ? [E: $i] :
              ( ( ( s @ ( fun @ A @ bool ) @ D )
                = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
              & ! [F: $i,G: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ F ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ G ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) )
                 => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ E ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ E ) @ ( s @ num @ F ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1690]) ).

thf(2162,axiom,
    ( ~ ( p @ ( s @ bool @ f0 ) )
  <=> ( p @ ( s @ bool @ t ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_CLAUSESu_WEAKu_conjunct1) ).

thf(16280,plain,
    ( ( ~ ( p @ ( s @ bool @ f0 ) )
     => ( p @ ( s @ bool @ t ) ) )
    & ( ( p @ ( s @ bool @ t ) )
     => ~ ( p @ ( s @ bool @ f0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2162]) ).

thf(2086,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ e_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ bool @ t ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_DEF) ).

thf(15827,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ e_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ! [C: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) )
            = ( s @ bool @ t ) ) )
      & ( ! [C: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) )
            = ( s @ bool @ t ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ e_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2086]) ).

thf(1899,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_RNEG) ).

thf(14729,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1899]) ).

thf(3048,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ! [B: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_REALu_INTERVALu_conjunct3) ).

thf(21072,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ! [B: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3048]) ).

thf(1054,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ) ) @ ( s @ int @ z ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ y ) ) ) @ ( s @ int @ z ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MINu_ACIu_conjunct1) ).

thf(9337,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ) ) @ ( s @ int @ z ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ y ) ) ) @ ( s @ int @ z ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1054]) ).

thf(2956,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ C ) ) ) )
    <=> ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ C ) ) ) ) ) )
        | ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_SUMu_THM) ).

thf(20534,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ C ) ) ) )
       => ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ C ) ) ) ) ) )
          | ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ C ) ) ) ) ) ) ) )
      & ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ C ) ) ) ) ) )
          | ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ C ) ) ) ) ) ) )
       => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2956]) ).

thf(884,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) )
          & ( ( s @ A @ C )
            = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ D ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMu_EXISTSu_EL) ).

thf(8447,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) )
            & ( ( s @ A @ C )
              = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ D ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) )
            & ( ( s @ A @ C )
              = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ D ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[884]) ).

thf(833,axiom,
    ! [A: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) )
        = ( s @ ( cart @ real @ A ) @ x ) )
    <=> ( ( s @ ( cart @ real @ A ) @ y )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_EQu_ADDR) ).

thf(8169,plain,
    ! [A: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) )
          = ( s @ ( cart @ real @ A ) @ x ) )
       => ( ( s @ ( cart @ real @ A ) @ y )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ y )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) )
          = ( s @ ( cart @ real @ A ) @ x ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[833]) ).

thf(335,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ ( prod @ A @ C ) @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( sum0 @ A @ B ) @ C ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ ( prod @ A @ C ) @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ ( sum0 @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ ( sum0 @ A @ B ) @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ ( sum0 @ A @ B ) @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ ( prod @ A @ C ) @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ ( prod @ A @ C ) @ ( prod @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ C ) @ bool ) @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ ( prod @ A @ C ) @ ( prod @ B @ C ) ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ ( prod @ A @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ A @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ A @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_RDISTRIB) ).

thf(5224,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ ( prod @ A @ C ) @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( sum0 @ A @ B ) @ C ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ ( prod @ A @ C ) @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ ( sum0 @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ ( sum0 @ A @ B ) @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ ( sum0 @ A @ B ) @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ ( prod @ A @ C ) @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ ( prod @ A @ C ) @ ( prod @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ C ) @ bool ) @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ ( prod @ A @ C ) @ ( prod @ B @ C ) ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ ( prod @ A @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ A @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ A @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).

thf(2978,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
          & ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ F ) ) ) ) )
             => ( ( s @ B @ E )
                = ( s @ B @ F ) ) ) )
    <=> ( ? [D: $i] :
          ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
        & ! [D: $i,E: $i] :
            ( ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
              & ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) ) )
           => ( ( s @ ( fun @ A @ B ) @ D )
              = ( s @ ( fun @ A @ B ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIQUEu_SKOLEMu_THM) ).

thf(20626,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
            & ! [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ F ) ) ) ) )
               => ( ( s @ B @ E )
                  = ( s @ B @ F ) ) ) )
       => ( ? [D: $i] :
            ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
          & ! [D: $i,E: $i] :
              ( ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
                & ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) ) )
             => ( ( s @ ( fun @ A @ B ) @ D )
                = ( s @ ( fun @ A @ B ) @ E ) ) ) ) )
      & ( ( ? [D: $i] :
            ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
          & ! [D: $i,E: $i] :
              ( ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
                & ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) ) )
             => ( ( s @ ( fun @ A @ B ) @ D )
                = ( s @ ( fun @ A @ B ) @ E ) ) ) )
       => ! [D: $i] :
            ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
            & ! [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ F ) ) ) ) )
               => ( ( s @ B @ E )
                  = ( s @ B @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2978]) ).

thf(2458,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_TRANSu_LESS) ).

thf(17847,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2458]) ).

thf(3070,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LTu_LE) ).

thf(21196,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3070]) ).

thf(1608,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTEu_ADD) ).

thf(12911,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1608]) ).

thf(1020,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EXPu_conjunct6) ).

thf(9191,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1020]) ).

thf(539,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
            <=> ? [H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ I ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ G ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ G ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_RESTRICTu_SET) ).

thf(6426,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
               => ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
              & ( ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ G ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).

thf(1504,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_NEGu_0) ).

thf(12261,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1504]) ).

thf(2600,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ F ) ) ) @ ( s @ B @ G ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ F ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ F ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SWAP) ).

thf(18635,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ F ) ) ) @ ( s @ B @ G ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ F ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ F ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2600]) ).

thf(1353,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIT0u_THM) ).

thf(11266,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1353]) ).

thf(1606,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHypermapo_THREE) ).

thf(12896,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1606]) ).

thf(201,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_LE) ).

thf(4326,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(2451,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) )
    <=> ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
       => ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_EXISTSu_IMPu_THM) ).

thf(17815,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ C ) ) )
       => ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) ) )
      & ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2451]) ).

thf(2585,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',acolumn) ).

thf(18546,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2585]) ).

thf(2274,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_LADDu_IMP) ).

thf(16845,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2274]) ).

thf(38,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ real ) ) @ B ) @ ( s @ real @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ real @ C ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ real ) ) @ B ) @ ( s @ real @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_CONST) ).

thf(3402,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ real ) ) @ B ) @ ( s @ real @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ real @ C ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ real ) ) @ B ) @ ( s @ real @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(2434,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_CROSS) ).

thf(17715,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2434]) ).

thf(2645,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
        & ( ( s @ A @ C )
         != ( s @ A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aless) ).

thf(18837,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
          & ( ( s @ A @ C )
           != ( s @ A @ D ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
          & ( ( s @ A @ C )
           != ( s @ A @ D ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2645]) ).

thf(1241,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_UNION) ).

thf(10463,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1241]) ).

thf(1440,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_BASIS) ).

thf(11881,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1440]) ).

thf(2851,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_MULu_SYM) ).

thf(19964,plain,
    ! [A: $i,B: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2851]) ).

thf(2031,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZRECSPACEu_RULESu_conjunct0) ).

thf(15579,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2031]) ).

thf(2153,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_UNIV) ).

thf(16229,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2153]) ).

thf(228,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_SUBSETu_SIMPLE) ).

thf(4487,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(1038,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                    = ( s @ A @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ num ) @ G ) ) ) @ ( s @ A @ H ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
             => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_IMAGEu_GEN) ).

thf(9268,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ num ) @ G ) ) ) @ ( s @ A @ H ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
             => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1038]) ).

thf(1644,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ! [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ H ) ) )
                  = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ I ) ) ) ) )
             => ( ( s @ B @ H )
                = ( s @ B @ I ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ C @ A ) @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_IMAGE) ).

thf(13202,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ! [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ H ) ) )
                  = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ I ) ) ) ) )
             => ( ( s @ B @ H )
                = ( s @ B @ I ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ C @ A ) @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1644]) ).

thf(2943,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( s @ A @ C )
      = ( s @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_REFL) ).

thf(20477,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( s @ A @ C )
      = ( s @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2943]) ).

thf(3137,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ empty ) )
      | ? [C: $i,D: $i] :
          ( ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSETu_CASES) ).

thf(21538,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ empty ) )
      | ? [C: $i,D: $i] :
          ( ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3137]) ).

thf(2918,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ! [G: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERS) ).

thf(20305,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                & ( ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                & ( ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2918]) ).

thf(395,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_RMUL) ).

thf(5589,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).

thf(724,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
          = ( s @ ( cart @ real @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBASISu_EXPANSION) ).

thf(7484,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
          = ( s @ ( cart @ real @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[724]) ).

thf(3044,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_DELETEu_EQ) ).

thf(21016,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
      & ( ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) )
       => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3044]) ).

thf(2651,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ B ) @ ( s @ ( list @ A ) @ nil ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ B ) @ ( s @ ( list @ A ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ B ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ B ) @ ( s @ ( list @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alistu_INDUCT) ).

thf(18861,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ B ) @ ( s @ ( list @ A ) @ nil ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ B ) @ ( s @ ( list @ A ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ B ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ B ) @ ( s @ ( list @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2651]) ).

thf(2393,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i,I: $i,J: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ G ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) )
     => ! [G: $i] :
          ( ! [H: $i,I: $i,J: $i] :
              ( ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
              = ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) )
         => ! [H: $i,I: $i,J: $i,K: $i,L: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ K ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_COMB) ).

thf(17411,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i,I: $i,J: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ G ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) )
     => ! [G: $i] :
          ( ! [H: $i,I: $i,J: $i] :
              ( ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
              = ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) )
         => ! [H: $i,I: $i,J: $i,K: $i,L: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ K ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ A ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2393]) ).

thf(2524,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCHAINu_SUBSET) ).

thf(18178,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2524]) ).

thf(630,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_LNEG) ).

thf(6971,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).

thf(1726,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_UNIQ) ).

thf(13660,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1726]) ).

thf(1873,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ B @ C ) @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ C @ A ) ) @ ( cart @ B @ C ) ) @ fstcart ) @ ( s @ ( cart @ B @ ( finite_sum @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( cart @ B @ ( finite_sum @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ ( cart @ B @ A ) @ ( cart @ B @ ( finite_sum @ C @ A ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ C ) @ D ) ) ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) ) )
      = ( s @ ( cart @ B @ C ) @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFSTCARTu_PASTECART) ).

thf(14617,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ B @ C ) @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ C @ A ) ) @ ( cart @ B @ C ) ) @ fstcart ) @ ( s @ ( cart @ B @ ( finite_sum @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( cart @ B @ ( finite_sum @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ ( cart @ B @ A ) @ ( cart @ B @ ( finite_sum @ C @ A ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ C ) @ D ) ) ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) ) )
      = ( s @ ( cart @ B @ C ) @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1873]) ).

thf(2720,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_LADDu_IMP) ).

thf(19230,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2720]) ).

thf(1294,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amatrixu_sub) ).

thf(10775,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1294]) ).

thf(1990,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUPu_BOUNDS) ).

thf(15357,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1990]) ).

thf(1639,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_NEG2) ).

thf(13187,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1639]) ).

thf(2591,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_RDISTRIB) ).

thf(18592,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2591]) ).

thf(310,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_SUBu_RZERO) ).

thf(5037,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(119,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_NEG) ).

thf(3794,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(836,axiom,
    ! [A: $i] :
      ( ( ( s @ num @ u_0 )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) )
    <=> ( ( s @ num @ u_0 )
        = ( s @ num @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EQu_conjunct4) ).

thf(8182,plain,
    ! [A: $i] :
      ( ( ( ( s @ num @ u_0 )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) )
       => ( ( s @ num @ u_0 )
          = ( s @ num @ A ) ) )
      & ( ( ( s @ num @ u_0 )
          = ( s @ num @ A ) )
       => ( ( s @ num @ u_0 )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[836]) ).

thf(1247,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_UNIONu_LZERO) ).

thf(10484,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1247]) ).

thf(1827,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
    <=> ~ ? [B: $i] :
          ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ B ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_DCHAIN) ).

thf(14347,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
       => ~ ? [B: $i] :
            ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ B ) @ ( s @ num @ C ) ) ) ) ) ) )
      & ( ~ ? [B: $i] :
            ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ B ) @ ( s @ num @ C ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1827]) ).

thf(1812,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_UNIONu_conjunct1) ).

thf(14264,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1812]) ).

thf(2732,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ i1 ) @ ( s @ A @ B ) ) )
      = ( s @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIu_DEF) ).

thf(19291,plain,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ i1 ) @ ( s @ A @ B ) ) )
      = ( s @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2732]) ).

thf(1091,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_SUCu_LE) ).

thf(9526,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1091]) ).

thf(2308,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_TRIANGLEu_LT) ).

thf(16982,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2308]) ).

thf(1240,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) )
        <=> ? [I: $i,J: $i] :
              ( ( ( p @ ( s @ bool @ J ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                    = ( s @ A @ G ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ H ) ) ) @ ( s @ A @ I ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ A @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ H ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ I ) ) ) ) )
             => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_GROUP) ).

thf(10433,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) )
           => ? [I: $i,J: $i] :
                ( ( ( p @ ( s @ bool @ J ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                      = ( s @ A @ G ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                      = ( s @ A @ G ) ) )
                 => ( p @ ( s @ bool @ J ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) ) )
          & ( ? [I: $i,J: $i] :
                ( ( ( p @ ( s @ bool @ J ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                      = ( s @ A @ G ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                      = ( s @ A @ G ) ) )
                 => ( p @ ( s @ bool @ J ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ H ) ) ) @ ( s @ A @ I ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ A @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ H ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ I ) ) ) ) )
             => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1240]) ).

thf(391,axiom,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ y ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ z ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ A @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_3u_conjunct0) ).

thf(5565,plain,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ y ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ z ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ A @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).

thf(2260,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
         => ( p @ ( s @ bool @ C ) ) )
    <=> ( ! [D: $i] : ( p @ ( s @ bool @ B ) )
       => ? [D: $i] : ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRIVu_EXISTSu_IMPu_THM) ).

thf(16777,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2260]) ).

thf(1598,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ A )
        = ( s @ num @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_OFu_NUMu_WELLDEF) ).

thf(12852,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ A )
        = ( s @ num @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1598]) ).

thf(2839,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIRWISEu_EMPTY) ).

thf(19900,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2839]) ).

thf(2208,axiom,
    ~ ( p @ ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSsrboolo_notu_falseu_isu_true) ).

thf(16513,plain,
    ~ ( p @ ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2208]) ).

thf(441,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_NUMSEG) ).

thf(5906,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).

thf(1729,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_PAIR) ).

thf(13675,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1729]) ).

thf(1786,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_UNIVu_conjunct0) ).

thf(14124,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1786]) ).

thf(1293,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_CARDu_IMP) ).

thf(10772,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1293]) ).

thf(1232,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) ) )
    <=> ( ( s @ ( list @ B ) @ l2 )
        = ( s @ ( list @ B ) @ nil ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALL2u_DEFu_conjunct0) ).

thf(10395,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) ) )
       => ( ( s @ ( list @ B ) @ l2 )
          = ( s @ ( list @ B ) @ nil ) ) )
      & ( ( ( s @ ( list @ B ) @ l2 )
          = ( s @ ( list @ B ) @ nil ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1232]) ).

thf(1855,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_COMPOSEu_NEG) ).

thf(14519,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1855]) ).

thf(1688,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LEu_conjunct5) ).

thf(13457,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1688]) ).

thf(575,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
        <=> ! [F: $i,G: $i] :
            ? [H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXTENDu_INSEG) ).

thf(6665,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
           => ! [F: $i,G: $i] :
              ? [H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
          & ( ! [F: $i,G: $i] :
              ? [H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).

thf(1225,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ D ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUPu_ASCLOSE) ).

thf(10358,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ D ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1225]) ).

thf(2915,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) )
    <=> ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
       => ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_FORALLu_IMPu_THM) ).

thf(20293,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ C ) ) )
       => ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) ) )
      & ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2915]) ).

thf(2419,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ C ) )
    <=> ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_EQ) ).

thf(17594,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) )
       => ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) )
      & ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2419]) ).

thf(1536,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
      <=> ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_0) ).

thf(12479,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
         => ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ empty ) ) )
        & ( ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ empty ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1536]) ).

thf(2044,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ A @ C ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_ID) ).

thf(15632,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ A @ C ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2044]) ).

thf(965,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
    <=> ? [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) ) )
          & ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
             => ( ? [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
                & ! [E: $i,F: $i] :
                    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ F ) ) ) ) ) ) )
                   => ( ( s @ A @ E )
                      = ( s @ A @ F ) ) ) ) )
          & ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
             => ( ? [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
                & ! [E: $i,F: $i] :
                    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
                   => ( ( s @ B @ E )
                      = ( s @ B @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_C) ).

thf(8879,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
       => ? [C: $i] :
            ( ! [D: $i,E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) ) )
            & ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
               => ( ? [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
                  & ! [E: $i,F: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ F ) ) ) ) ) ) )
                     => ( ( s @ A @ E )
                        = ( s @ A @ F ) ) ) ) )
            & ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
               => ( ? [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
                  & ! [E: $i,F: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
                     => ( ( s @ B @ E )
                        = ( s @ B @ F ) ) ) ) ) ) )
      & ( ? [C: $i] :
            ( ! [D: $i,E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) ) )
            & ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
               => ( ? [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
                  & ! [E: $i,F: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ D ) ) ) @ ( s @ A @ F ) ) ) ) ) ) )
                     => ( ( s @ A @ E )
                        = ( s @ A @ F ) ) ) ) )
            & ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
               => ( ? [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
                  & ! [E: $i,F: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ C ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
                     => ( ( s @ B @ E )
                        = ( s @ B @ F ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[965]) ).

thf(2642,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ empty ) @ ( s @ A @ B ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEMPTY) ).

thf(18824,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ empty ) @ ( s @ A @ B ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2642]) ).

thf(1814,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIFFu_EQu_EMPTY) ).

thf(14273,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1814]) ).

thf(1654,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_DOUBLE) ).

thf(13276,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1654]) ).

thf(1893,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIMINDEXu_UNIV) ).

thf(14694,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1893]) ).

thf(3076,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_SUB) ).

thf(21229,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3076]) ).

thf(1885,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_LT) ).

thf(14665,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1885]) ).

thf(1898,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_NEG2) ).

thf(14726,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1898]) ).

thf(2656,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_LEu_MULu_RCANCELu_IMP) ).

thf(18897,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2656]) ).

thf(2923,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ F ) ) ) ) )
     => ! [C: $i] :
          ( ( ! [D: $i] :
                ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ D ) ) ) )
            & ! [D: $i,E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ F ) ) ) ) )
            & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_FINITE) ).

thf(20352,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ F ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ( ! [D: $i] :
                ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ D ) ) ) )
            & ! [D: $i,E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ F ) ) ) ) )
            & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2923]) ).

thf(922,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) @ ( s @ num @ H ) ) )
          = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ H ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ H ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ G ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) ) ) @ ( s @ num @ I ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ G ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i,J: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) )
             => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ J ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ J ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_VSUMu_PARTIALu_PRE) ).

thf(8667,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) @ ( s @ num @ H ) ) )
          = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ H ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ H ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ G ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) ) ) @ ( s @ num @ I ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ G ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i,J: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) )
             => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ J ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ J ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[922]) ).

thf(1969,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_CLAUSESu_conjunct2) ).

thf(15217,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1969]) ).

thf(2163,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_LT) ).

thf(16283,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2163]) ).

thf(2661,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_SUB) ).

thf(18921,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2661]) ).

thf(2103,axiom,
    ? [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) @ ( s @ ind0 @ C ) ) )
            = ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) @ ( s @ ind0 @ D ) ) ) )
        <=> ( ( s @ ind0 @ C )
            = ( s @ ind0 @ D ) ) )
      & ! [C: $i] :
          ( ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) @ ( s @ ind0 @ C ) ) )
         != ( s @ ind0 @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINDu_SUCu_0u_EXISTS) ).

thf(15948,plain,
    ? [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) @ ( s @ ind0 @ C ) ) )
              = ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) @ ( s @ ind0 @ D ) ) ) )
           => ( ( s @ ind0 @ C )
              = ( s @ ind0 @ D ) ) )
          & ( ( ( s @ ind0 @ C )
              = ( s @ ind0 @ D ) )
           => ( ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) @ ( s @ ind0 @ C ) ) )
              = ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) @ ( s @ ind0 @ D ) ) ) ) ) )
      & ! [C: $i] :
          ( ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) @ ( s @ ind0 @ C ) ) )
         != ( s @ ind0 @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2103]) ).

thf(574,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
    <=> ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
        | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_POSu_LT) ).

thf(6655,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
       => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).

thf(2821,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( prod @ A @ B ) @ C ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( prod @ A @ B ) @ C ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ ( prod @ A @ B ) @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ ( prod @ A @ B ) @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_MULu_ASSOC) ).

thf(19787,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( prod @ A @ B ) @ C ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( prod @ A @ B ) @ C ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ ( prod @ A @ B ) @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ ( prod @ A @ B ) @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2821]) ).

thf(704,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ ( cart @ real @ n10 ) @ B )
          = ( s @ ( cart @ real @ n10 ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ n10 ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ONE) ).

thf(7345,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ ( cart @ real @ n10 ) @ B )
          = ( s @ ( cart @ real @ n10 ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ n10 ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[704]) ).

thf(858,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ p1 ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_ACu_conjunct1) ).

thf(8298,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ p1 ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[858]) ).

thf(3159,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) @ ( s @ ( list @ B ) @ H ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) @ ( s @ ( list @ B ) @ H ) ) ) ) )
            <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) @ ( s @ ( list @ B ) @ H ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aANDu_ALL2) ).

thf(21627,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) @ ( s @ ( list @ B ) @ H ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) @ ( s @ ( list @ B ) @ H ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) @ ( s @ ( list @ B ) @ H ) ) ) ) )
              & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) @ ( s @ ( list @ B ) @ H ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) @ ( s @ ( list @ B ) @ H ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) @ ( s @ ( list @ B ) @ H ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3159]) ).

thf(78,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_NSUM) ).

thf(3616,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(555,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_CAUCHYu_SCHWARZu_DIV) ).

thf(6515,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).

thf(608,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ card ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_NUMSEG) ).

thf(6851,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ card ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).

thf(116,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
      <=> ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUPu_LTu_FINITE) ).

thf(3781,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
         => ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
        & ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(1960,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ C ) ) ) ) ) )
     => ! [C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ q0 ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
        <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aANDu_ALL) ).

thf(15175,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ C ) ) ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ C ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) )
     => ! [C: $i] :
          ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ q0 ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
          & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ q0 ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1960]) ).

thf(1026,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ B ) ) )
          = ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ C ) ) ) )
      <=> ( ( s @ num @ B )
          = ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_INDEXu_INJ) ).

thf(9218,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( ( ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ B ) ) )
            = ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ C ) ) ) )
         => ( ( s @ num @ B )
            = ( s @ num @ C ) ) )
        & ( ( ( s @ num @ B )
            = ( s @ num @ C ) )
         => ( ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ B ) ) )
            = ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1026]) ).

thf(1580,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
        = ( s @ bool @ f0 ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
      <=> ( ( ( s @ A @ C )
            = ( s @ A @ B ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSeq2o_MEM) ).

thf(12725,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
        = ( s @ bool @ f0 ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
       => ( ( ( s @ A @ C )
            = ( s @ A @ B ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
      & ( ( ( ( s @ A @ C )
            = ( s @ A @ B ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1580]) ).

thf(1932,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
       => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_IMPu_FORALLu_THM) ).

thf(15034,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( p @ ( s @ bool @ B ) )
         => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ( p @ ( s @ bool @ B ) )
         => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1932]) ).

thf(983,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( ( s @ num @ C )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_MONOu_LT) ).

thf(9000,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          & ( ( s @ num @ C )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          & ( ( s @ num @ C )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[983]) ).

thf(477,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_DOUBLE) ).

thf(6078,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).

thf(2095,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_IMAGEu_GEN) ).

thf(15885,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2095]) ).

thf(1796,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
         => ( p @ ( s @ bool @ C ) ) )
    <=> ( ? [D: $i] : ( p @ ( s @ bool @ B ) )
       => ! [D: $i] : ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRIVu_FORALLu_IMPu_THM) ).

thf(14149,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1796]) ).

thf(1061,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ real ) @ neutral ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNEUTRALu_REALu_ADD) ).

thf(9356,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ real ) @ neutral ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1061]) ).

thf(3107,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_ELIMu_THMu_conjunct1) ).

thf(21403,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) ) )
          & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3107]) ).

thf(2183,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ A ) )
        | ( p @ ( s @ bool @ B ) ) )
    <=> ! [C: $i] :
          ( ( ( p @ ( s @ bool @ A ) )
           => ( p @ ( s @ bool @ C ) ) )
         => ( ( ( p @ ( s @ bool @ B ) )
             => ( p @ ( s @ bool @ C ) ) )
           => ( p @ ( s @ bool @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORu_DEF) ).

thf(16376,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ B ) ) )
       => ! [C: $i] :
            ( ( ( p @ ( s @ bool @ A ) )
             => ( p @ ( s @ bool @ C ) ) )
           => ( ( ( p @ ( s @ bool @ B ) )
               => ( p @ ( s @ bool @ C ) ) )
             => ( p @ ( s @ bool @ C ) ) ) ) )
      & ( ! [C: $i] :
            ( ( ( p @ ( s @ bool @ A ) )
             => ( p @ ( s @ bool @ C ) ) )
           => ( ( ( p @ ( s @ bool @ B ) )
               => ( p @ ( s @ bool @ C ) ) )
             => ( p @ ( s @ bool @ C ) ) ) )
       => ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2183]) ).

thf(1770,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) ) ) @ a_a_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ int @ bool ) ) ) @ intu_mod ) @ ( s @ int @ C ) ) ) ) ) )
    <=> ? [D: $i] :
          ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_congruent) ).

thf(13960,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) ) ) @ a_a_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ int @ bool ) ) ) @ intu_mod ) @ ( s @ int @ C ) ) ) ) ) )
       => ? [D: $i] :
            ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
            = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) )
      & ( ? [D: $i] :
            ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
            = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ ( fun @ int @ ( fun @ int @ bool ) ) @ bool ) ) ) @ a_a_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ int @ bool ) ) ) @ intu_mod ) @ ( s @ int @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1770]) ).

thf(567,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ( ( s @ A @ C )
                  = ( s @ A @ E ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSINGu_GSPECu_conjunct1) ).

thf(6551,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( s @ A @ C )
                    = ( s @ A @ E ) ) )
                & ( ( ( s @ A @ C )
                    = ( s @ A @ E ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( s @ A @ C )
                    = ( s @ A @ E ) ) )
                & ( ( ( s @ A @ C )
                    = ( s @ A @ E ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).

thf(56,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ s0 ) ) ) )
         => ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ t0 ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ s0 ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ t0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_MONO) ).

thf(3481,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ s0 ) ) ) )
         => ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ t0 ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ s0 ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ t0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(1285,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_LZERO) ).

thf(10726,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1285]) ).

thf(68,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) )
        = ( s @ ( list @ A ) @ nil ) )
    <=> ( ( s @ ( list @ B ) @ D )
        = ( s @ ( list @ B ) @ nil ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_EQu_NIL) ).

thf(3562,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) )
          = ( s @ ( list @ A ) @ nil ) )
       => ( ( s @ ( list @ B ) @ D )
          = ( s @ ( list @ B ) @ nil ) ) )
      & ( ( ( s @ ( list @ B ) @ D )
          = ( s @ ( list @ B ) @ nil ) )
       => ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) )
          = ( s @ ( list @ A ) @ nil ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(638,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTEu_ADD2) ).

thf(6996,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[638]) ).

thf(650,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GEu_conjunct8) ).

thf(7068,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[650]) ).

thf(1935,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_TOTAL) ).

thf(15051,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1935]) ).

thf(1662,axiom,
    ! [A: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ s0 )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ s0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ s0 ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUPu_LE) ).

thf(13298,plain,
    ! [A: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ s0 )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ s0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ s0 ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1662]) ).

thf(2619,axiom,
    ! [A: $i] :
      ( ( ( ( s @ bool @ t )
          = ( s @ bool @ A ) )
      <=> ( p @ ( s @ bool @ A ) ) )
      & ( ( ( s @ bool @ A )
          = ( s @ bool @ t ) )
      <=> ( p @ ( s @ bool @ A ) ) )
      & ( ( ( s @ bool @ f0 )
          = ( s @ bool @ A ) )
      <=> ~ ( p @ ( s @ bool @ A ) ) )
      & ( ( ( s @ bool @ A )
          = ( s @ bool @ f0 ) )
      <=> ~ ( p @ ( s @ bool @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_CLAUSES) ).

thf(18729,plain,
    ! [A: $i] :
      ( ( ( ( s @ bool @ t )
          = ( s @ bool @ A ) )
       => ( p @ ( s @ bool @ A ) ) )
      & ( ( p @ ( s @ bool @ A ) )
       => ( ( s @ bool @ t )
          = ( s @ bool @ A ) ) )
      & ( ( ( s @ bool @ A )
          = ( s @ bool @ t ) )
       => ( p @ ( s @ bool @ A ) ) )
      & ( ( p @ ( s @ bool @ A ) )
       => ( ( s @ bool @ A )
          = ( s @ bool @ t ) ) )
      & ( ( ( s @ bool @ f0 )
          = ( s @ bool @ A ) )
       => ~ ( p @ ( s @ bool @ A ) ) )
      & ( ~ ( p @ ( s @ bool @ A ) )
       => ( ( s @ bool @ f0 )
          = ( s @ bool @ A ) ) )
      & ( ( ( s @ bool @ A )
          = ( s @ bool @ f0 ) )
       => ~ ( p @ ( s @ bool @ A ) ) )
      & ( ~ ( p @ ( s @ bool @ A ) )
       => ( ( s @ bool @ A )
          = ( s @ bool @ f0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2619]) ).

thf(721,axiom,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ h0 ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFILTERu_conjunct1) ).

thf(7452,plain,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ h0 ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[721]) ).

thf(2412,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_CROSS) ).

thf(17577,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2412]) ).

thf(953,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHypermapo_GEu_1) ).

thf(8838,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[953]) ).

thf(2513,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_SUB) ).

thf(18133,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2513]) ).

thf(2388,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) )
            <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALL2u_ANDu_RIGHT) ).

thf(17375,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ) )
              & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2388]) ).

thf(598,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_SQUAREu_NORM) ).

thf(6804,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).

thf(2340,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) )
     => ( ! [C: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) ) ) @ ( s @ A @ C ) ) )
            = ( s @ A @ C ) )
        & ! [C: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) ) )
            = ( s @ A @ C ) )
        & ! [C: $i,D: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOIDALu_AC) ).

thf(17126,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) )
     => ( ! [C: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) ) ) @ ( s @ A @ C ) ) )
            = ( s @ A @ C ) )
        & ! [C: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) ) )
            = ( s @ A @ C ) )
        & ! [C: $i,D: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2340]) ).

thf(2303,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_div) ).

thf(16965,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2303]) ).

thf(840,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_GT0) ).

thf(8204,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[840]) ).

thf(3038,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_gt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_gt) ).

thf(20986,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_gt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3038]) ).

thf(1967,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ numleft ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ bool @ A ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numright ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSUMu_DEST) ).

thf(15207,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ numleft ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ bool @ A ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numright ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ num @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1967]) ).

thf(3052,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) )
     => ( ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) ) )
        = ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADJOINTu_ADJOINT) ).

thf(21096,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) )
     => ( ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) ) )
        = ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3052]) ).

thf(1415,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
          = ( s @ real @ a0 ) )
      <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ a0 ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
          & ! [B: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ a0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPu_UNIQUEu_FINITE) ).

thf(11756,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
            = ( s @ real @ a0 ) )
         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ a0 ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
            & ! [B: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ a0 ) ) ) ) ) ) )
        & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ a0 ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
            & ! [B: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ a0 ) ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
            = ( s @ real @ a0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1415]) ).

thf(2636,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_REFL) ).

thf(18800,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2636]) ).

thf(1870,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ B ) )
       => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_FORALLu_IMPu_THM) ).

thf(14591,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ( p @ ( s @ bool @ B ) )
         => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ B ) )
         => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1870]) ).

thf(766,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
      = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_CLAUSESu_conjunct0) ).

thf(7748,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
      = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[766]) ).

thf(139,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_01) ).

thf(3895,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(1829,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ D ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ E ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_VSUM) ).

thf(14354,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ D ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ E ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1829]) ).

thf(3023,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_EMPTYu_conjunct1) ).

thf(20885,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3023]) ).

thf(2147,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ G ) ) ) )
    <=> ? [H: $i,I: $i,J: $i] :
          ( ! [K: $i,L: $i,M: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ M ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ M ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ L ) ) ) ) ) ) )
          & ! [K: $i,L: $i,M: $i] :
              ( ! [N: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ N ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ M ) ) ) ) ) )
                 => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ K ) @ ( s @ A @ N ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ L ) @ ( s @ A @ N ) ) ) ) )
             => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ M ) ) )
                  = ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ L ) ) ) @ ( s @ C @ M ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ M ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ L ) ) ) @ ( s @ C @ M ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ J ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ M ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ J ) @ ( s @ ( fun @ A @ B ) @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) )
          & ! [K: $i,L: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ F ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ G ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ K ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ J ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',atailadmissible) ).

thf(16164,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ G ) ) ) )
       => ? [H: $i,I: $i,J: $i] :
            ( ! [K: $i,L: $i,M: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ M ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ M ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ L ) ) ) ) ) ) )
            & ! [K: $i,L: $i,M: $i] :
                ( ! [N: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ N ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ M ) ) ) ) ) )
                   => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ K ) @ ( s @ A @ N ) ) )
                      = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ L ) @ ( s @ A @ N ) ) ) ) )
               => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ M ) ) )
                    = ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ L ) ) ) @ ( s @ C @ M ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ M ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ L ) ) ) @ ( s @ C @ M ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ J ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ M ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ J ) @ ( s @ ( fun @ A @ B ) @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) )
            & ! [K: $i,L: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ F ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) )
               => ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ G ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ K ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ J ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) ) ) )
      & ( ? [H: $i,I: $i,J: $i] :
            ( ! [K: $i,L: $i,M: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ M ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ M ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ L ) ) ) ) ) ) )
            & ! [K: $i,L: $i,M: $i] :
                ( ! [N: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ N ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ M ) ) ) ) ) )
                   => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ K ) @ ( s @ A @ N ) ) )
                      = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ L ) @ ( s @ A @ N ) ) ) ) )
               => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ M ) ) )
                    = ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ L ) ) ) @ ( s @ C @ M ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ M ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ L ) ) ) @ ( s @ C @ M ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ J ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ M ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ J ) @ ( s @ ( fun @ A @ B ) @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) )
            & ! [K: $i,L: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ F ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) )
               => ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ G ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ H ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ K ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ A ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ J ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ B ) ) @ G ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2147]) ).

thf(2592,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_INTERu_ABSORPTION) ).

thf(18595,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ B ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ B ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2592]) ).

thf(2860,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_LDISTRIB) ).

thf(20011,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2860]) ).

thf(2073,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
     => ? [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ num @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINRECu_EXISTSu_LEMMA) ).

thf(15735,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
     => ? [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ num @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2073]) ).

thf(860,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_ADDu_C) ).

thf(8303,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[860]) ).

thf(1224,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
    <=> ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aorthogonal) ).

thf(10351,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1224]) ).

thf(110,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_SQUAREu_ABS) ).

thf(3766,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(826,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_UNION) ).

thf(8133,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[826]) ).

thf(1349,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_MUL2) ).

thf(11250,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1349]) ).

thf(2058,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_RMUL) ).

thf(15683,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2058]) ).

thf(1339,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_3) ).

thf(11171,plain,
    ! [A: $i] :
      ( ( ! [B: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
       => ! [B: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1339]) ).

thf(1531,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_BOUNDu_GEN) ).

thf(12447,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1531]) ).

thf(666,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ min ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_MODu_EXPu_MIN) ).

thf(7168,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ min ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[666]) ).

thf(2863,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_NEGATION) ).

thf(20034,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2863]) ).

thf(958,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ A @ D ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINSEGu_FL) ).

thf(8851,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ A @ D ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[958]) ).

thf(3121,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ B ) )
       => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_EXISTSu_IMPu_THM) ).

thf(21467,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ( p @ ( s @ bool @ B ) )
         => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ B ) )
         => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3121]) ).

thf(2295,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_IMAGE) ).

thf(16933,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2295]) ).

thf(271,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUCu_SUB1) ).

thf(4792,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(1686,axiom,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ x ) ) ) @ ( s @ nadd @ y ) ) ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ x ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ y ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ahrealu_addu_th) ).

thf(13452,plain,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ x ) ) ) @ ( s @ nadd @ y ) ) ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ x ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ y ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1686]) ).

thf(2202,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aGT) ).

thf(16491,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2202]) ).

thf(957,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_RCANCELu_IMP) ).

thf(8848,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[957]) ).

thf(1718,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_minu_th) ).

thf(13628,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1718]) ).

thf(2960,axiom,
    ! [A: $i,B: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_UNIVu_PSUBSET) ).

thf(20556,plain,
    ! [A: $i,B: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2960]) ).

thf(1373,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_GTu_SQUARE) ).

thf(11436,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1373]) ).

thf(2399,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ D ) ) ) )
    <=> ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ D ) ) ) @ ( s @ ( cart @ B @ C ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_PASTECART) ).

thf(17492,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ D ) ) ) )
       => ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ D ) ) ) @ ( s @ ( cart @ B @ C ) @ E ) ) ) ) ) ) )
      & ( ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ D ) ) ) @ ( s @ ( cart @ B @ C ) @ E ) ) ) ) ) )
       => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2399]) ).

thf(1003,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ int @ B )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SOSu_EQu_0) ).

thf(9064,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ int @ B )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      & ( ( ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ int @ B )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1003]) ).

thf(2251,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injf ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ B ) ) )
        = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injf ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ C ) ) ) )
    <=> ( ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ B )
        = ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJFu_INJ) ).

thf(16733,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injf ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ B ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injf ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ C ) ) ) )
       => ( ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ B )
          = ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ C ) ) )
      & ( ( ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ B )
          = ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ C ) )
       => ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injf ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ B ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injf ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2251]) ).

thf(1610,axiom,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GEu_conjunct3) ).

thf(12916,plain,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1610]) ).

thf(144,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_LT0) ).

thf(3915,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(513,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ADDu_conjunct1) ).

thf(6268,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).

thf(2193,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTEu_TRANS) ).

thf(16456,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2193]) ).

thf(1172,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ B ) @ D ) ) ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) )
        <=> ! [F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ num @ F ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ D ) @ ( s @ num @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( cart @ B @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ B ) @ D ) ) )
          = ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( cart @ B @ A ) ) @ h_ ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alambda) ).

thf(10022,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ B ) @ D ) ) ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) )
           => ! [F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
               => ( ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ num @ F ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ D ) @ ( s @ num @ F ) ) ) ) ) )
          & ( ! [F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
               => ( ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ num @ F ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ D ) @ ( s @ num @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ B ) @ D ) ) ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( cart @ B @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ B ) @ D ) ) )
          = ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( cart @ B @ A ) ) @ h_ ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1172]) ).

thf(2167,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_UNIONS) ).

thf(16304,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2167]) ).

thf(2962,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
          & ( p @ ( s @ bool @ C ) ) )
    <=> ( ? [D: $i] : ( p @ ( s @ bool @ B ) )
        & ? [D: $i] : ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRIVu_EXISTSu_ANDu_THM) ).

thf(20569,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2962]) ).

thf(1266,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
        <=> ? [G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ G ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ E ) ) ) )
            & ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ G ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_PRODUCTu_DEPENDENT) ).

thf(10616,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
           => ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ G ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ G ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
          & ( ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ G ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ G ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ E ) ) ) )
            & ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1266]) ).

thf(1387,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
    <=> ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( s @ int @ B )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
        | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_POSu_LE) ).

thf(11542,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
       => ( ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ int @ B )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ int @ B )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1387]) ).

thf(231,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ n ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_SINGu_NUMSEG) ).

thf(4496,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ n ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(1965,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_SUC) ).

thf(15202,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1965]) ).

thf(1310,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( ( s @ num @ C )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_MONOu_LTu_IMP) ).

thf(10908,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( ( s @ num @ C )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1310]) ).

thf(1245,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_SUBu_DISTRIB) ).

thf(10474,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1245]) ).

thf(903,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_POWu_2) ).

thf(8539,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[903]) ).

thf(1847,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
      & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWO) ).

thf(14483,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
      & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1847]) ).

thf(2226,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_0) ).

thf(16623,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2226]) ).

thf(2882,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_ACIu_conjunct0) ).

thf(20166,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2882]) ).

thf(2644,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ bool @ t ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALLu_T) ).

thf(18834,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ bool @ t ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2644]) ).

thf(2069,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTEu_TOTAL) ).

thf(15723,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2069]) ).

thf(1405,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_SUBu_LDISTRIB) ).

thf(11699,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1405]) ).

thf(2500,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LTu_ADD) ).

thf(18040,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2500]) ).

thf(991,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_POSu_EQu_0) ).

thf(9027,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[991]) ).

thf(114,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_IMAGEu_LE) ).

thf(3777,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(1069,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D )
           != ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) )
         => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIFFu_UNIONSu_NONEMPTY) ).

thf(9379,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D )
           != ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) )
         => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1069]) ).

thf(1586,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ADDu_conjunct2) ).

thf(12757,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1586]) ).

thf(1680,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_REFL) ).

thf(13392,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1680]) ).

thf(1010,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ B )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ real @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_DIVu_LMUL) ).

thf(9159,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ B )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ real @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1010]) ).

thf(1182,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_POWu_L1) ).

thf(10078,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1182]) ).

thf(2678,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) )
      = ( s @ ( fun @ B @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPPORTu_CLAUSESu_conjunct0) ).

thf(18992,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) )
      = ( s @ ( fun @ B @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2678]) ).

thf(1166,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
    <=> ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSsrnato_leqE) ).

thf(9994,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1166]) ).

thf(32,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
    <=> ? [C: $i] :
          ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_EXISTS) ).

thf(3308,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
       => ? [C: $i] :
            ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) )
      & ( ? [C: $i] :
            ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(1562,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_EXP) ).

thf(12646,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1562]) ).

thf(181,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ B ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_DIFFSu_ALT) ).

thf(4124,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ B ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(698,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_ADD2u_REV) ).

thf(7326,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[698]) ).

thf(1865,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_UNIONu_IMP) ).

thf(14559,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1865]) ).

thf(315,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) )
            <=> ? [F: $i,G: $i] :
                  ( ( ( p @ ( s @ bool @ G ) )
                  <=> ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) )
                      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ real @ F ) ) ) ) ) )
         => ! [C: $i,D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) ) ) ) )
            <=> ? [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
                  & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ E ) ) )
                   != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POLYFUNu_FINITEu_ROOTS) ).

thf(5056,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) )
               => ? [F: $i,G: $i] :
                    ( ( ( p @ ( s @ bool @ G ) )
                     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) )
                        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
                    & ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) )
                        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
                     => ( p @ ( s @ bool @ G ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ real @ F ) ) ) ) ) )
              & ( ? [F: $i,G: $i] :
                    ( ( ( p @ ( s @ bool @ G ) )
                     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) )
                        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
                    & ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) )
                        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
                     => ( p @ ( s @ bool @ G ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ real @ F ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) ) ) )
         => ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) ) ) ) )
               => ? [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
                    & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ E ) ) )
                     != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
              & ( ? [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
                    & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ E ) ) )
                     != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).

thf(418,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFINITEu_CARDu_LE) ).

thf(5770,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).

thf(2567,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_MIN) ).

thf(18420,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2567]) ).

thf(963,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) )
        <=> ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ e_ ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) ) )
        <=> ! [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_PAIREDu_THM) ).

thf(8866,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) )
           => ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
          & ( ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ e_ ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) ) )
           => ! [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) )
          & ( ! [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ e_ ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[963]) ).

thf(923,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ n10 ) @ ( cart @ real @ n10 ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ n10 ) @ ( cart @ real @ n10 ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( prod @ ( cart @ real @ n10 ) @ ( cart @ real @ n10 ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ ( cart @ real @ n10 ) @ ( prod @ ( cart @ real @ n10 ) @ ( cart @ real @ n10 ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ n10 ) @ A ) ) ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_REAL) ).

thf(8671,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ n10 ) @ ( cart @ real @ n10 ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ n10 ) @ ( cart @ real @ n10 ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( prod @ ( cart @ real @ n10 ) @ ( cart @ real @ n10 ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ ( cart @ real @ n10 ) @ ( prod @ ( cart @ real @ n10 ) @ ( cart @ real @ n10 ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ n10 ) @ A ) ) ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[923]) ).

thf(2666,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_MULu_SYM) ).

thf(18938,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2666]) ).

thf(3108,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_NORM) ).

thf(21415,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3108]) ).

thf(1709,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_DIFFSu_ALT) ).

thf(13571,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1709]) ).

thf(3013,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_MINIMAL) ).

thf(20845,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3013]) ).

thf(351,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ x ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ x ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_invu_th) ).

thf(5301,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ x ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ x ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).

thf(3141,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_CLAUSESu_conjunct0) ).

thf(21560,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3141]) ).

thf(1161,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ( ( s @ A @ C )
           != ( s @ A @ D ) ) )
     => ! [C: $i] :
          ( ! [D: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ D ) ) ) )
         => ! [D: $i] :
              ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
              = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_SETu_OFu_LIST) ).

thf(9971,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ( ( s @ A @ C )
             != ( s @ A @ D ) ) )
          & ( ( ( s @ A @ C )
             != ( s @ A @ D ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ D ) ) ) )
         => ! [D: $i] :
              ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
              = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1161]) ).

thf(2532,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_NEG) ).

thf(18252,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2532]) ).

thf(171,axiom,
    ( ( ( p @ ( s @ bool @ p1 ) )
      | ( p @ ( s @ bool @ q ) )
      | ( p @ ( s @ bool @ r0 ) ) )
  <=> ( ( p @ ( s @ bool @ p1 ) )
      | ( p @ ( s @ bool @ q ) )
      | ( p @ ( s @ bool @ r0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJu_ACIu_conjunct1) ).

thf(4030,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(2510,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_INFINITEu_CONG) ).

thf(18120,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2510]) ).

thf(2733,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_ADD) ).

thf(19294,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2733]) ).

thf(864,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LTu_conjunct2) ).

thf(8330,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[864]) ).

thf(2307,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) ) ) ) ) @ ( s @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORDINALu_UNIONu_LEMMA) ).

thf(16979,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) ) ) ) ) @ ( s @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2307]) ).

thf(1673,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_conjunct0) ).

thf(13361,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1673]) ).

thf(1882,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_RADD) ).

thf(14644,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1882]) ).

thf(1443,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_ofu_numu_th) ).

thf(11912,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1443]) ).

thf(897,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_SUBSETu_EQ) ).

thf(8504,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[897]) ).

thf(769,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ C ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_CONSTu_NUMSEG) ).

thf(7761,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ C ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[769]) ).

thf(3080,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_IMPu_LE) ).

thf(21239,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3080]) ).

thf(1430,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ C @ H ) ) ) )
        <=> ? [I: $i,J: $i,K: $i] :
              ( ( ( p @ ( s @ bool @ K ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) @ gspec ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_PRODUCTu_DEPENDENT) ).

thf(11831,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ C @ H ) ) ) )
           => ? [I: $i,J: $i,K: $i] :
                ( ( ( p @ ( s @ bool @ K ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ K ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) )
          & ( ? [I: $i,J: $i,K: $i] :
                ( ( ( p @ ( s @ bool @ K ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ K ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) @ gspec ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1430]) ).

thf(1762,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ D )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_LTu_ALL) ).

thf(13923,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ D )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1762]) ).

thf(986,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_MONO) ).

thf(9013,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[986]) ).

thf(2180,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ? [I: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) )
                    = ( s @ B @ H ) ) )
              & ! [I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) )
                      = ( s @ B @ H ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ J ) ) )
                      = ( s @ B @ H ) ) )
                 => ( ( s @ A @ I )
                    = ( s @ A @ J ) ) ) ) )
        & ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ H ) ) ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ E ) @ ( s @ A @ H ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_EQu_GENERAL) ).

thf(16351,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ? [I: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) )
                    = ( s @ B @ H ) ) )
              & ! [I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) )
                      = ( s @ B @ H ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ J ) ) )
                      = ( s @ B @ H ) ) )
                 => ( ( s @ A @ I )
                    = ( s @ A @ J ) ) ) ) )
        & ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ H ) ) ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ E ) @ ( s @ A @ H ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2180]) ).

thf(2466,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_INTER) ).

thf(17888,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2466]) ).

thf(494,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_LSQRT) ).

thf(6182,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).

thf(2937,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ univ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIVu_SUBSET) ).

thf(20441,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ univ ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ univ ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2937]) ).

thf(2617,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_LDISTRIB) ).

thf(18717,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2617]) ).

thf(2135,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum )
      = ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) ) @ iterate ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asum) ).

thf(16118,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum )
      = ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) ) @ iterate ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2135]) ).

thf(2356,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
         => ( ( s @ A @ C )
            = ( s @ A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPOSETu_ANTISYM) ).

thf(17197,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
         => ( ( s @ A @ C )
            = ( s @ A @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2356]) ).

thf(1267,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_ADDR) ).

thf(10633,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1267]) ).

thf(3057,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) ) ) )
        & ! [G: $i,H: $i,I: $i] :
            ( ! [J: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ J ) ) ) @ ( s @ B @ I ) ) ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ J ) ) ) ) )
           => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ I ) ) )
                = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ B @ I ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ I ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ B @ I ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ F ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ I ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ F ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) )
        & ! [G: $i,H: $i,I: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ I ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ I ) ) ) @ ( s @ B @ H ) ) ) ) )
        & ! [G: $i,H: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ F ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
     => ? [G: $i] :
        ! [H: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ H ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_RECu_TAILu_GENERALi_) ).

thf(21124,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) ) ) )
        & ! [G: $i,H: $i,I: $i] :
            ( ! [J: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ J ) ) ) @ ( s @ B @ I ) ) ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ J ) ) ) ) )
           => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ I ) ) )
                = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ B @ I ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ I ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ B @ I ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ F ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ I ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ F ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) )
        & ! [G: $i,H: $i,I: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ I ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ I ) ) ) @ ( s @ B @ H ) ) ) ) )
        & ! [G: $i,H: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ F ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
     => ? [G: $i] :
        ! [H: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ H ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3057]) ).

thf(807,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
        <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ G ) ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ F ) ) ) )
            & ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ H ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_UNIONSu_LE) ).

thf(7993,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
           => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ G ) ) ) ) ) ) )
          & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ F ) ) ) )
            & ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ H ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[807]) ).

thf(1626,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POLYu_CLAUSESu_conjunct8) ).

thf(13039,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1626]) ).

thf(2474,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_LE) ).

thf(17925,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2474]) ).

thf(581,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_INV) ).

thf(6707,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).

thf(753,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_MUL) ).

thf(7701,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[753]) ).

thf(3134,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SING) ).

thf(21530,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3134]) ).

thf(1595,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUBu_conjunct0) ).

thf(12843,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1595]) ).

thf(1501,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_MUL2) ).

thf(12254,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1501]) ).

thf(1157,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_MOD) ).

thf(9960,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1157]) ).

thf(728,axiom,
    ! [A: $i] :
      ( ( ! [B: $i,C: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) ) )
        & ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_POSu_MONOu_1) ).

thf(7496,plain,
    ! [A: $i] :
      ( ( ! [B: $i,C: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) ) )
        & ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[728]) ).

thf(127,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_MODu_REFL) ).

thf(3820,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(2569,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_SQUAREu_INFINITE) ).

thf(18437,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2569]) ).

thf(3059,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ columns ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ rows ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOLUMNSu_TRANSP) ).

thf(21143,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ columns ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ rows ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3059]) ).

thf(8,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) ) @ map2 ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h1 ) ) ) @ ( s @ ( list @ B ) @ t1 ) ) ) ) ) @ ( s @ ( list @ C ) @ l0 ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) @ ( s @ B @ h1 ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( list @ C ) @ C ) @ hd ) @ ( s @ ( list @ C ) @ l0 ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) ) @ map2 ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) ) ) @ ( s @ ( list @ B ) @ t1 ) ) ) @ ( s @ ( list @ C ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ C ) ) @ tl ) @ ( s @ ( list @ C ) @ l0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAP2u_DEFu_conjunct1) ).

thf(3204,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) ) @ map2 ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h1 ) ) ) @ ( s @ ( list @ B ) @ t1 ) ) ) ) ) @ ( s @ ( list @ C ) @ l0 ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) @ ( s @ B @ h1 ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( list @ C ) @ C ) @ hd ) @ ( s @ ( list @ C ) @ l0 ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) ) @ map2 ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) ) ) @ ( s @ ( list @ B ) @ t1 ) ) ) @ ( s @ ( list @ C ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ C ) ) @ tl ) @ ( s @ ( list @ C ) @ l0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(2990,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) )
           => ? [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) @ ( s @ A @ H ) ) ) ) )
        & ! [F: $i,G: $i,H: $i,I: $i,J: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ A @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ B @ H ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ A @ G ) ) ) ) )
           => ( ( ( s @ C @ I )
                = ( s @ C @ J ) )
              & ( ( s @ A @ F )
                = ( s @ A @ G ) ) ) ) )
     => ? [F: $i] :
        ! [G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ G ) ) ) )
         => ( ? [I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) @ ( s @ A @ I ) ) ) )
          <=> ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ F ) @ ( s @ B @ G ) ) )
              = ( s @ C @ H ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINRECu_FUNu_LEMMA) ).

thf(20707,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) )
           => ? [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) @ ( s @ A @ H ) ) ) ) )
        & ! [F: $i,G: $i,H: $i,I: $i,J: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ A @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ B @ H ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ A @ G ) ) ) ) )
           => ( ( ( s @ C @ I )
                = ( s @ C @ J ) )
              & ( ( s @ A @ F )
                = ( s @ A @ G ) ) ) ) )
     => ? [F: $i] :
        ! [G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ G ) ) ) )
         => ( ( ? [I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) @ ( s @ A @ I ) ) ) )
             => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ F ) @ ( s @ B @ G ) ) )
                = ( s @ C @ H ) ) )
            & ( ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ F ) @ ( s @ B @ G ) ) )
                = ( s @ C @ H ) )
             => ? [I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) @ ( s @ A @ I ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2990]) ).

thf(48,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ B ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_ADDu_RCANCELu_0) ).

thf(3442,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ B ) )
       => ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(2620,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
    ! [F: $i,G: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apairu_RECURSION) ).

thf(18754,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
    ! [F: $i,G: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2620]) ).

thf(416,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ b0 ) ) ) @ ( s @ real @ c ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ b0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ c ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCollectu_geomo_PERu_MUL3u_conjunct0) ).

thf(5765,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ b0 ) ) ) @ ( s @ real @ c ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ b0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ c ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).

thf(1045,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_RDIVu_EQ) ).

thf(9300,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1045]) ).

thf(2725,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_SING) ).

thf(19267,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2725]) ).

thf(241,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i,D: $i] :
    ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ALTMUL) ).

thf(4546,plain,
    ! [A: $i,B: $i] :
    ? [C: $i,D: $i] :
    ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(2173,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSEGu_SUBSET) ).

thf(16332,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2173]) ).

thf(366,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_0) ).

thf(5429,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ A )
          = ( s @ real @ B ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ B ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).

thf(2227,axiom,
    ( ( p @ ( s @ bool @ f0 ) )
  <=> ! [A: $i] : ( p @ ( s @ bool @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFu_DEF) ).

thf(16626,plain,
    ( ( ( p @ ( s @ bool @ f0 ) )
     => ! [A: $i] : ( p @ ( s @ bool @ A ) ) )
    & ( ! [A: $i] : ( p @ ( s @ bool @ A ) )
     => ( p @ ( s @ bool @ f0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2227]) ).

thf(1165,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) )
      <=> ( ( s @ real @ A )
          = ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_INJ) ).

thf(9989,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) )
         => ( ( s @ real @ A )
            = ( s @ real @ B ) ) )
        & ( ( ( s @ real @ A )
            = ( s @ real @ B ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1165]) ).

thf(2888,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_CARDu_MUL) ).

thf(20191,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2888]) ).

thf(2622,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPOSETu_TRANS) ).

thf(18760,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2622]) ).

thf(5,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) ) @ map2 ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h1 ) ) ) @ ( s @ ( list @ B ) @ t1 ) ) ) ) ) @ ( s @ ( list @ C ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ C ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( list @ C ) @ ( list @ C ) ) ) @ cons ) @ ( s @ C @ h2 ) ) ) @ ( s @ ( list @ C ) @ t2 ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) @ ( s @ B @ h1 ) ) ) @ ( s @ C @ h2 ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) ) @ map2 ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) ) ) @ ( s @ ( list @ B ) @ t1 ) ) ) @ ( s @ ( list @ C ) @ t2 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAP2u_conjunct1) ).

thf(3195,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) ) @ map2 ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h1 ) ) ) @ ( s @ ( list @ B ) @ t1 ) ) ) ) ) @ ( s @ ( list @ C ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ C ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( list @ C ) @ ( list @ C ) ) ) @ cons ) @ ( s @ C @ h2 ) ) ) @ ( s @ ( list @ C ) @ t2 ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) @ ( s @ B @ h1 ) ) ) @ ( s @ C @ h2 ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) ) @ map2 ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ f ) ) ) @ ( s @ ( list @ B ) @ t1 ) ) ) @ ( s @ ( list @ C ) @ t2 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(2337,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) )
         => ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_MEASUREu_GEN) ).

thf(17117,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) )
         => ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2337]) ).

thf(1041,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_SUM) ).

thf(9287,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1041]) ).

thf(1461,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
      <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ r0 ) @ ( s @ A @ x ) ) ) @ ( s @ B @ D ) ) ) ) )
      & ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) ) @ u_match ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ u_seqpattern ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ r0 ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ s0 ) ) ) ) )
        = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) ) @ u_match ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ r0 ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) ) @ u_match ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ s0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATCHu_SEQPATTERN) ).

thf(11976,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
       => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ r0 ) @ ( s @ A @ x ) ) ) @ ( s @ B @ D ) ) ) ) )
      & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ r0 ) @ ( s @ A @ x ) ) ) @ ( s @ B @ D ) ) ) )
       => ( p @ ( s @ bool @ C ) ) )
      & ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) ) @ u_match ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ u_seqpattern ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ r0 ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ s0 ) ) ) ) )
        = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) ) @ u_match ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ r0 ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) ) @ u_match ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ s0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1461]) ).

thf(2957,axiom,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ A ) @ ( s @ hreal @ B ) ) ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ A ) @ ( s @ hreal @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) ) )
     => ? [B: $i] :
          ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ A ) @ ( s @ hreal @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) )
          & ! [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ A ) @ ( s @ hreal @ D ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ D ) ) ) @ ( s @ hreal @ C ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_COMPLETE) ).

thf(20541,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ A ) @ ( s @ hreal @ B ) ) ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ A ) @ ( s @ hreal @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) ) )
     => ? [B: $i] :
          ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ A ) @ ( s @ hreal @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) )
          & ! [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ A ) @ ( s @ hreal @ D ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ D ) ) ) @ ( s @ hreal @ C ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2957]) ).

thf(85,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ F ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) ) @ o ) @ ( s @ ( fun @ B @ real ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_IMAGEu_LE) ).

thf(3644,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ F ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) ) @ o ) @ ( s @ ( fun @ B @ real ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(283,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
    ? [F: $i] :
      ( ( ( p @ ( s @ bool @ F ) )
      <=> ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) ) )
      & ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
        = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPPORTu_CLAUSESu_conjunct1) ).

thf(4896,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
    ? [F: $i] :
      ( ( ( p @ ( s @ bool @ F ) )
       => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) ) )
      & ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) )
       => ( p @ ( s @ bool @ F ) ) )
      & ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
        = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(2422,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
        <=> ? [G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ! [I: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ I ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
            <=> ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERSu_GSPECu_conjunct0) ).

thf(17610,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
           => ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ! [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ I ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ! [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ I ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
          & ( ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ! [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ I ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ! [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ I ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
               => ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) )
              & ( ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2422]) ).

thf(471,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ card ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_NUMSEGu_1) ).

thf(6039,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ card ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).

thf(1263,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_ADD) ).

thf(10587,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1263]) ).

thf(1742,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        | ( ( s @ num @ B )
          = ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_MULTu_LCANCEL) ).

thf(13803,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          | ( ( s @ num @ B )
            = ( s @ num @ C ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          | ( ( s @ num @ B )
            = ( s @ num @ C ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1742]) ).

thf(1074,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SUBu_NUMSEG) ).

thf(9410,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1074]) ).

thf(562,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_MULu_EQu_conjunct0) ).

thf(6538,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).

thf(877,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ r0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIRWISEu_conjunct0) ).

thf(8426,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ r0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[877]) ).

thf(1186,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_RDIVu_EQ) ).

thf(10089,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1186]) ).

thf(1632,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ y ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ y ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_ACu_conjunct0) ).

thf(13091,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ y ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ y ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1632]) ).

thf(2919,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_SUBu_RADD) ).

thf(20340,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2919]) ).

thf(2746,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_TRANS) ).

thf(19342,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2746]) ).

thf(565,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ p1 ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_ACu_conjunct1) ).

thf(6547,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ p1 ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).

thf(1290,axiom,
    ! [A: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) )
        = ( s @ num @ u_0 ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ u_0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EQu_conjunct2) ).

thf(10751,plain,
    ! [A: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ u_0 ) )
       => ( ( s @ num @ A )
          = ( s @ num @ u_0 ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ u_0 ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1290]) ).

thf(1025,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_LINV) ).

thf(9216,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1025]) ).

thf(1096,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LEu_conjunct4) ).

thf(9561,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1096]) ).

thf(1006,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_RREC) ).

thf(9084,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1006]) ).

thf(750,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUCu_conjunct3) ).

thf(7683,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[750]) ).

thf(866,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ u ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ u ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_INSERT) ).

thf(8357,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ u ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ u ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[866]) ).

thf(2697,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) ) ) )
     => ! [C: $i,D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ! [H: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ H ) ) ) @ ( s @ B @ G ) ) ) )
                 => ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ H ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ H ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) ) ) ) ) ) )
             => ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ G ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) )
         => ? [E: $i] :
            ! [F: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_RECu_INVARIANT) ).

thf(19108,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) ) ) )
     => ! [C: $i,D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ! [H: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ H ) ) ) @ ( s @ B @ G ) ) ) )
                 => ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ H ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ H ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) ) ) ) ) ) )
             => ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ G ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) )
         => ? [E: $i] :
            ! [F: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2697]) ).

thf(1849,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALLu_FILTER) ).

thf(14489,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1849]) ).

thf(2441,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSET) ).

thf(17751,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2441]) ).

thf(1324,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIT0) ).

thf(11010,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1324]) ).

thf(538,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_RDISTRIB) ).

thf(6423,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).

thf(476,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ? [E: $i] :
          ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
          & ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
             => ? [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) )
                    = ( s @ A @ F ) ) ) )
          & ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ B @ F )
                = ( s @ B @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_BIJECTION) ).

thf(6066,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ? [E: $i] :
          ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
          & ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
             => ? [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) )
                    = ( s @ A @ F ) ) ) )
          & ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ B @ F )
                = ( s @ B @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).

thf(1457,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_TRIANGLESu_LE) ).

thf(11966,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1457]) ).

thf(349,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ C ) ) ) )
        <=> ! [D: $i,E: $i] :
            ? [F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ c0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ l0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ C ) @ ( s @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ bool @ F ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aKLu_POSETu_LEMMA) ).

thf(5274,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ C ) ) ) )
           => ! [D: $i,E: $i] :
              ? [F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ c0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ l0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ c0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ l0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ C ) @ ( s @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ bool @ F ) ) ) ) ) )
          & ( ! [D: $i,E: $i] :
              ? [F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ c0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ l0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ c0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ l0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ C ) @ ( s @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ bool @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).

thf(1191,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_COMBINEu_L) ).

thf(10142,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1191]) ).

thf(1205,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_SUPERSET) ).

thf(10221,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1205]) ).

thf(801,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( s @ real @ B )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ENTIRE) ).

thf(7945,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      & ( ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[801]) ).

thf(2093,axiom,
    ( ( s @ bool @ t )
    = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTacticsu_jordano_unifyu_existsu_tacu_example) ).

thf(15881,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2093]) ).

thf(783,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GTu_conjunct7) ).

thf(7877,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[783]) ).

thf(1063,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_LNEG) ).

thf(9360,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1063]) ).

thf(900,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
        | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_NUMSEG) ).

thf(8522,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[900]) ).

thf(1336,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ bool ) @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
            & ! [E: $i,F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
                  & ( ( s @ ( fun @ A @ bool ) @ E )
                   != ( s @ ( fun @ A @ bool ) @ F ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
               => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ G ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ bool ) @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_UNIONSu_NONZERO) ).

thf(11111,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ bool ) @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
            & ! [E: $i,F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
                  & ( ( s @ ( fun @ A @ bool ) @ E )
                   != ( s @ ( fun @ A @ bool ) @ F ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
               => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ G ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ bool ) @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1336]) ).

thf(2630,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_PSUBSETu_TRANS) ).

thf(18782,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2630]) ).

thf(2989,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPOSETu_FLEQ) ).

thf(20704,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2989]) ).

thf(66,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ F ) ) ) )
        <=> ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ s0 ) @ ( s @ A @ F ) ) )
            = ( s @ D @ x ) ) )
     => ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
          <=> ? [G: $i] :
                ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ s0 ) @ ( s @ A @ G ) ) )
                = ( s @ D @ x ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( fun @ C @ ( fun @ D @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) ) ) @ cons ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ ( fun @ C @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ D ) @ s0 ) ) ) @ ( s @ ( fun @ C @ ( fun @ A @ B ) ) @ t0 ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ clauses ) ) ) ) ) @ ( s @ C @ f ) ) ) @ ( s @ D @ x ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ B ) ) @ t0 ) @ ( s @ C @ f ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( fun @ C @ ( fun @ D @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ clauses ) ) ) @ ( s @ C @ f ) ) ) @ ( s @ D @ x ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCASEWISEu_conjunct1) ).

thf(3542,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ F ) ) ) )
           => ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ s0 ) @ ( s @ A @ F ) ) )
              = ( s @ D @ x ) ) )
          & ( ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ s0 ) @ ( s @ A @ F ) ) )
              = ( s @ D @ x ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
           => ? [G: $i] :
                ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ s0 ) @ ( s @ A @ G ) ) )
                = ( s @ D @ x ) ) )
          & ( ? [G: $i] :
                ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ s0 ) @ ( s @ A @ G ) ) )
                = ( s @ D @ x ) )
           => ( p @ ( s @ bool @ F ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( fun @ C @ ( fun @ D @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) ) ) @ cons ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ ( fun @ C @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ D ) @ s0 ) ) ) @ ( s @ ( fun @ C @ ( fun @ A @ B ) ) @ t0 ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ clauses ) ) ) ) ) @ ( s @ C @ f ) ) ) @ ( s @ D @ x ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ B ) ) @ t0 ) @ ( s @ C @ f ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( fun @ C @ ( fun @ D @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ clauses ) ) ) @ ( s @ C @ f ) ) ) @ ( s @ D @ x ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(193,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_UNIONu_GEN) ).

thf(4275,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(207,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',avsum) ).

thf(4349,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(819,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_RADD) ).

thf(8069,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[819]) ).

thf(1049,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ D ) @ ( s @ B @ F ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_NORMu_BOUND) ).

thf(9315,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ D ) @ ( s @ B @ F ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1049]) ).

thf(1426,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ E ) ) ) @ ( s @ real @ F ) ) ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_CLOSED) ).

thf(11812,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ E ) ) ) @ ( s @ real @ F ) ) ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ B ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1426]) ).

thf(1170,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_SUB) ).

thf(10010,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1170]) ).

thf(2498,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
    <=> ( ( s @ num @ B )
        = ( s @ num @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_ADDu_LCANCEL) ).

thf(18030,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
       => ( ( s @ num @ B )
          = ( s @ num @ C ) ) )
      & ( ( ( s @ num @ B )
          = ( s @ num @ C ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2498]) ).

thf(484,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ! [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ H ) ) ) ) )
               => ( ( s @ A @ G )
                  = ( s @ A @ H ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_IMAGEu_INJu_GENERAL) ).

thf(6113,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ! [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ H ) ) ) ) )
               => ( ( s @ A @ G )
                  = ( s @ A @ H ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).

thf(1220,axiom,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y1 ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y2 ) ) ) ) )
   => ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y2 ) ) ) )
    <=> ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y2 ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRATu_LEMMA5) ).

thf(10337,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y1 ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y2 ) ) ) ) )
   => ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y2 ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y2 ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y1 ) ) ) ) )
      & ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y2 ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y1 ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y2 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1220]) ).

thf(401,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POSu_NZ) ).

thf(5640,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).

thf(1553,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_SIGN2) ).

thf(12606,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1553]) ).

thf(1124,axiom,
    ( ( ( ( p @ ( s @ bool @ b ) )
       => ( p @ ( s @ bool @ a ) ) )
      & ( ( p @ ( s @ bool @ c0 ) )
       => ( p @ ( s @ bool @ d0 ) ) ) )
   => ( ( ( p @ ( s @ bool @ a ) )
       => ( p @ ( s @ bool @ c0 ) ) )
     => ( ( p @ ( s @ bool @ b ) )
       => ( p @ ( s @ bool @ d0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOu_IMP) ).

thf(9805,plain,
    ( ( ( ( p @ ( s @ bool @ b ) )
       => ( p @ ( s @ bool @ a ) ) )
      & ( ( p @ ( s @ bool @ c0 ) )
       => ( p @ ( s @ bool @ d0 ) ) ) )
   => ( ( ( p @ ( s @ bool @ a ) )
       => ( p @ ( s @ bool @ c0 ) ) )
     => ( ( p @ ( s @ bool @ b ) )
       => ( p @ ( s @ bool @ d0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1124]) ).

thf(461,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_1u_LE) ).

thf(5989,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).

thf(523,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ? [D: $i] :
        ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLOWERu_BOUNDu_FINITEu_SET) ).

thf(6332,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ? [D: $i] :
        ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).

thf(1653,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_POSu_LE) ).

thf(13273,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1653]) ).

thf(2769,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_gt) ).

thf(19525,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2769]) ).

thf(2806,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_CART) ).

thf(19710,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2806]) ).

thf(222,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_BETWEEN2) ).

thf(4428,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(765,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
      <=> ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
          = ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_RDIVu_EQ) ).

thf(7743,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
            = ( s @ real @ B ) ) )
        & ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
            = ( s @ real @ B ) )
         => ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[765]) ).

thf(502,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
    <=> ( ( ( s @ A @ x )
          = ( s @ A @ h0 ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMu_conjunct1) ).

thf(6208,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
       => ( ( ( s @ A @ x )
            = ( s @ A @ h0 ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
      & ( ( ( ( s @ A @ x )
            = ( s @ A @ h0 ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).

thf(1600,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ B ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLISTu_OFu_SETu_PROPERTIES) ).

thf(12869,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ B ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1600]) ).

thf(429,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
      <=> ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUPu_LEu_FINITE) ).

thf(5836,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
         => ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
        & ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).

thf(1806,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) )
      = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( prod @ A @ B ) ) @ absu_prod ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOMMAu_DEF) ).

thf(14228,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) )
      = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( prod @ A @ B ) ) @ absu_prod ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1806]) ).

thf(2951,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_INu_INSERT) ).

thf(20508,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          & ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          & ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) )
       => ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2951]) ).

thf(2934,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ E ) ) ) ) )
         => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ F ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ F ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_EQu_NUMSEG) ).

thf(20418,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ E ) ) ) ) )
         => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ F ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ F ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2934]) ).

thf(3084,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_ADD2) ).

thf(21268,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3084]) ).

thf(641,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( s @ real @ B )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_DIVu_EQu_0) ).

thf(7011,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      & ( ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[641]) ).

thf(2898,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFINITEu_SUPERSET) ).

thf(20232,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2898]) ).

thf(3074,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORDINALu_UNION) ).

thf(21220,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3074]) ).

thf(1462,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_1) ).

thf(11981,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1462]) ).

thf(12,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ A @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( list @ A ) ) ) @ replicate ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) ) )
      = ( s @ num @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLENGTHu_REPLICATE) ).

thf(3216,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ A @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( list @ A ) ) ) @ replicate ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) ) )
      = ( s @ num @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(2252,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numfst ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ num @ A ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numsnd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMPAIRu_DEST) ).

thf(16740,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numfst ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ num @ A ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numsnd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ num @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2252]) ).

thf(1848,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_REFL) ).

thf(14487,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1848]) ).

thf(768,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) )
    <=> ( p @ ( s @ bool @ f0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EQu_conjunct8) ).

thf(7754,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) )
       => ( p @ ( s @ bool @ f0 ) ) )
      & ( ( p @ ( s @ bool @ f0 ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[768]) ).

thf(3104,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ D ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) ) )
     => ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ D ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSYMMETRICu_LINEARu_IMAGE) ).

thf(21394,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ D ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) ) )
     => ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ D ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3104]) ).

thf(3100,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ A ) ) ) )
    <=> ( ( ( s @ ind0 @ A )
          = ( s @ ind0 @ indu_0 ) )
        | ? [B: $i] :
            ( ( ( s @ ind0 @ A )
              = ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMu_REPu_CASES) ).

thf(21367,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ A ) ) ) )
       => ( ( ( s @ ind0 @ A )
            = ( s @ ind0 @ indu_0 ) )
          | ? [B: $i] :
              ( ( ( s @ ind0 @ A )
                = ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ B ) ) ) ) ) ) )
      & ( ( ( ( s @ ind0 @ A )
            = ( s @ ind0 @ indu_0 ) )
          | ? [B: $i] :
              ( ( ( s @ ind0 @ A )
                = ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ B ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3100]) ).

thf(2687,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i,I: $i,J: $i,K: $i] :
          ( ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) )
          = ( s @ E @ ( i @ ( s @ ( fun @ E @ E ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ E ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ E @ ( fun @ E @ E ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ G ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) )
     => ! [G: $i] :
          ( ! [H: $i,I: $i,J: $i,K: $i] :
              ( ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) )
              = ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) ) )
         => ! [H: $i] :
              ( ! [I: $i,J: $i,K: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) )
                <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) )
             => ! [I: $i] :
                  ( ! [J: $i,K: $i] :
                      ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) )
                      = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) ) )
                 => ! [J: $i] :
                      ( ! [K: $i,L: $i,M: $i,N: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) )
                        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ K ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) ) ) )
                     => ! [K: $i] :
                          ( ! [L: $i,M: $i,N: $i] :
                              ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) )
                              = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) )
                         => ! [L: $i,M: $i,N: $i,O: $i,P: $i,Q: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) ) ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ P ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) ) ) ) ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) ) ) ) )
                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_COND) ).

thf(19052,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i,I: $i,J: $i,K: $i] :
          ( ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) )
          = ( s @ E @ ( i @ ( s @ ( fun @ E @ E ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ E ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ E @ ( fun @ E @ E ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ G ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) )
     => ! [G: $i] :
          ( ! [H: $i,I: $i,J: $i,K: $i] :
              ( ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) )
              = ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) ) )
         => ! [H: $i] :
              ( ! [I: $i,J: $i,K: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) )
                   => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) )
                  & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) )
             => ! [I: $i] :
                  ( ! [J: $i,K: $i] :
                      ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) )
                      = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) ) )
                 => ! [J: $i] :
                      ( ! [K: $i,L: $i,M: $i,N: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) )
                           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ K ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) ) ) )
                          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ K ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) ) )
                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) ) ) )
                     => ! [K: $i] :
                          ( ! [L: $i,M: $i,N: $i] :
                              ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) )
                              = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) )
                         => ! [L: $i,M: $i,N: $i,O: $i,P: $i,Q: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) ) ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ P ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) ) ) ) ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) ) ) ) )
                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2687]) ).

thf(2184,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_EQ) ).

thf(16391,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2184]) ).

thf(2085,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ! [C: $i] :
          ( ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
             => ( p @ ( s @ bool @ C ) ) )
         => ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_DEF) ).

thf(15819,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ! [C: $i] :
            ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
               => ( p @ ( s @ bool @ C ) ) )
           => ( p @ ( s @ bool @ C ) ) ) )
      & ( ! [C: $i] :
            ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
               => ( p @ ( s @ bool @ C ) ) )
           => ( p @ ( s @ bool @ C ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2085]) ).

thf(2820,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
        = ( s @ int @ B ) )
    <=> ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_EQ) ).

thf(19780,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
          = ( s @ int @ B ) )
       => ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      & ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
          = ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2820]) ).

thf(357,axiom,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ m ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ C ) ) ) ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ m ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ m ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ m ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ m ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSEQu_MONOu_LEMMA) ).

thf(5338,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ m ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ C ) ) ) ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ m ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ m ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ m ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ m ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).

thf(2038,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
        | ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_ORu_FORALLu_THM) ).

thf(15601,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( p @ ( s @ bool @ B ) )
          | ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ( p @ ( s @ bool @ B ) )
          | ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2038]) ).

thf(2558,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ C ) ) )
        = ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ D ) ) ) )
    <=> ( ( s @ B @ C )
        = ( s @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asumu_INJECTIVEu_conjunct1) ).

thf(18380,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ C ) ) )
          = ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ D ) ) ) )
       => ( ( s @ B @ C )
          = ( s @ B @ D ) ) )
      & ( ( ( s @ B @ C )
          = ( s @ B @ D ) )
       => ( ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ C ) ) )
          = ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2558]) ).

thf(2781,axiom,
    ( ( ! [A: $i,B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
   => ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_WLOGu_LE) ).

thf(19589,plain,
    ( ( ! [A: $i,B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
   => ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2781]) ).

thf(2255,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ A @ C ) )
     => ! [C: $i] :
          ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ A @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) )
          = ( s @ ( list @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_ID) ).

thf(16757,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ A @ C ) )
     => ! [C: $i] :
          ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ A @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) )
          = ( s @ ( list @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2255]) ).

thf(2332,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_ADDu_ASSOC) ).

thf(17097,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2332]) ).

thf(1361,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ D ) ) ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) )
        <=> ? [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ D ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
            <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) ) ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) ) )
         => ! [E: $i] :
              ( ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) ) )
              = ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSETu_PAIRu_THM) ).

thf(11331,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ D ) ) ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) )
           => ? [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ D ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
          & ( ? [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ D ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ D ) ) ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
               => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) ) ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) ) )
              & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) ) ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) ) ) )
         => ! [E: $i] :
              ( ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) ) )
              = ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1361]) ).

thf(1968,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_RNEG) ).

thf(15214,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1968]) ).

thf(3126,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_BOUNDS) ).

thf(21491,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3126]) ).

thf(1512,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ m ) ) ) @ ( s @ real @ n ) ) ) ) ) @ ( s @ real @ p1 ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ m ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ n ) ) ) @ ( s @ real @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_ACu_conjunct1) ).

thf(12291,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ m ) ) ) @ ( s @ real @ n ) ) ) ) ) @ ( s @ real @ p1 ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ m ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ n ) ) ) @ ( s @ real @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1512]) ).

thf(1081,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
       != ( s @ ( fun @ A @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_UNIONu_OVERLAP) ).

thf(9480,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
       != ( s @ ( fun @ A @ bool ) @ empty ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1081]) ).

thf(1930,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SINGu_NUMSEG) ).

thf(15028,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1930]) ).

thf(2796,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_TRANS) ).

thf(19658,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2796]) ).

thf(2033,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ A )
        = ( s @ num @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_IMPu_LE) ).

thf(15586,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ A )
        = ( s @ num @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2033]) ).

thf(614,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ B @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSETu_OFu_LISTu_MAP) ).

thf(6905,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ B @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).

thf(1845,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ D ) ) ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) )
        <=> ( ( ( ( s @ bool @ D )
                = ( s @ bool @ t ) )
             => ( ( s @ A @ F )
                = ( s @ A @ C ) ) )
            & ( ( ( s @ bool @ D )
                = ( s @ bool @ f0 ) )
             => ( ( s @ A @ F )
                = ( s @ A @ E ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONDu_DEF) ).

thf(14455,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ D ) ) ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) )
           => ( ( ( ( s @ bool @ D )
                  = ( s @ bool @ t ) )
               => ( ( s @ A @ F )
                  = ( s @ A @ C ) ) )
              & ( ( ( s @ bool @ D )
                  = ( s @ bool @ f0 ) )
               => ( ( s @ A @ F )
                  = ( s @ A @ E ) ) ) ) )
          & ( ( ( ( ( s @ bool @ D )
                  = ( s @ bool @ t ) )
               => ( ( s @ A @ F )
                  = ( s @ A @ C ) ) )
              & ( ( ( s @ bool @ D )
                  = ( s @ bool @ f0 ) )
               => ( ( s @ A @ F )
                  = ( s @ A @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ D ) ) ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1845]) ).

thf(2007,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( cart @ A @ B ) @ bool ) @ univ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_CARTu_UNIV) ).

thf(15477,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( cart @ A @ B ) @ bool ) @ univ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2007]) ).

thf(2924,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ E ) ) ) )
             => ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ F ) ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ B ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ num @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ B ) @ ( s @ ( fun @ num @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) )
     => ? [C: $i] :
        ! [D: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ B ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_RECu_num) ).

thf(20364,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ E ) ) ) )
             => ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ F ) ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ B ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ num @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ B ) @ ( s @ ( fun @ num @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) )
     => ? [C: $i] :
        ! [D: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ B ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2924]) ).

thf(2258,axiom,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ? [B: $i] :
          ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
          & ! [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ D ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_COMPLETE) ).

thf(16772,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ? [B: $i] :
          ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
          & ! [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ D ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2258]) ).

thf(2537,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ u_destu_rec ) @ ( s @ ( recspace @ A ) @ B ) ) )
        = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ u_destu_rec ) @ ( s @ ( recspace @ A ) @ C ) ) ) )
    <=> ( ( s @ ( recspace @ A ) @ B )
        = ( s @ ( recspace @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDESTu_RECu_INJ) ).

thf(18292,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ u_destu_rec ) @ ( s @ ( recspace @ A ) @ B ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ u_destu_rec ) @ ( s @ ( recspace @ A ) @ C ) ) ) )
       => ( ( s @ ( recspace @ A ) @ B )
          = ( s @ ( recspace @ A ) @ C ) ) )
      & ( ( ( s @ ( recspace @ A ) @ B )
          = ( s @ ( recspace @ A ) @ C ) )
       => ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ u_destu_rec ) @ ( s @ ( recspace @ A ) @ B ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ u_destu_rec ) @ ( s @ ( recspace @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2537]) ).

thf(1358,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) )
     => ? [B: $i] :
          ( ! [C: $i,D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ D ) ) ) ) ) ) )
          & ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFINITEu_ENUMERATE) ).

thf(11321,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) )
     => ? [B: $i] :
          ( ! [C: $i,D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ D ) ) ) ) ) ) )
          & ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1358]) ).

thf(3119,axiom,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_ANTISYM) ).

thf(21447,plain,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3119]) ).

thf(675,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_NORMu_ALLSUBSETSu_BOUND) ).

thf(7209,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[675]) ).

thf(2076,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_ADD2) ).

thf(15742,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2076]) ).

thf(1389,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_PREu_conjunct0) ).

thf(11564,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1389]) ).

thf(187,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
        & ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( ( s @ int @ B )
        = ( s @ int @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_EQ) ).

thf(4179,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
        & ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( ( s @ int @ B )
        = ( s @ int @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(2124,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ D ) ) )
          = ( s @ bool @ f0 ) )
     => ! [D: $i,E: $i] :
        ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
          <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
              & ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( ( s @ B @ G )
                    = ( s @ B @ H ) ) ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ B ) ) @ u_function ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ A @ E ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ B ) @ h_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ B ) @ h_ ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',au_FUNCTION) ).

thf(16035,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ D ) ) )
          = ( s @ bool @ f0 ) )
     => ! [D: $i,E: $i] :
        ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
              & ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( ( s @ B @ G )
                    = ( s @ B @ H ) ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
              & ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( ( s @ B @ G )
                    = ( s @ B @ H ) ) ) )
           => ( p @ ( s @ bool @ F ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ B ) ) @ u_function ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ A @ E ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ B ) @ h_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ B ) @ h_ ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2124]) ).

thf(1780,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_EQu_LDOT) ).

thf(14103,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) )
       => ! [D: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1780]) ).

thf(1712,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_gt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_GT) ).

thf(13580,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_gt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1712]) ).

thf(890,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( ( p @ ( s @ bool @ E ) )
      <=> ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aELu_CONS) ).

thf(8468,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( ( p @ ( s @ bool @ E ) )
       => ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ E ) ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).

thf(2606,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ G ) ) ) )
         => ? [H: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ H ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ G ) ) ) ) )
    <=> ? [G: $i] :
        ! [H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ H ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ H ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ G ) @ ( s @ C @ H ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFUNCTIONu_FACTORSu_RIGHTu_GEN) ).

thf(18659,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ G ) ) ) )
           => ? [H: $i] :
                ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ H ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ G ) ) ) ) )
       => ? [G: $i] :
          ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ H ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ H ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ G ) @ ( s @ C @ H ) ) ) ) ) ) ) )
      & ( ? [G: $i] :
          ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ H ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ H ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ G ) @ ( s @ C @ H ) ) ) ) ) ) )
       => ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ G ) ) ) )
           => ? [H: $i] :
                ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ H ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2606]) ).

thf(2281,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ A ) )
        & ( p @ ( s @ bool @ B ) ) )
    <=> ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ C ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ C ) @ ( s @ bool @ t ) ) ) @ ( s @ bool @ t ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aANDu_DEF) ).

thf(16873,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ B ) ) )
       => ! [C: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ C ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ C ) @ ( s @ bool @ t ) ) ) @ ( s @ bool @ t ) ) ) ) )
      & ( ! [C: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ C ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ C ) @ ( s @ bool @ t ) ) ) @ ( s @ bool @ t ) ) ) )
       => ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2281]) ).

thf(3139,axiom,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLETu_ANTISYM) ).

thf(21550,plain,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3139]) ).

thf(246,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) ) )
        <=> ? [C: $i,D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ y ) ) ) @ ( s @ nadd @ D ) ) ) ) ) )
     => ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ x ) ) ) @ ( s @ hreal @ y ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ahrealu_add) ).

thf(4622,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) ) )
           => ? [C: $i,D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ y ) ) ) @ ( s @ nadd @ D ) ) ) ) ) )
          & ( ? [C: $i,D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ y ) ) ) @ ( s @ nadd @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) ) ) ) )
     => ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ x ) ) ) @ ( s @ hreal @ y ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(1367,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ D ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( finite_image @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ ( finite_image @ B ) @ A ) ) @ destu_cart ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ ( finite_image @ B ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ B ) ) @ finiteu_index ) @ ( s @ num @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',afiniteu_index) ).

thf(11378,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ D ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( finite_image @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ ( finite_image @ B ) @ A ) ) @ destu_cart ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ ( finite_image @ B ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ B ) ) @ finiteu_index ) @ ( s @ num @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1367]) ).

thf(515,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ B )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ? [C: $i] :
          ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_INFu_SUBSET) ).

thf(6272,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ B )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ? [C: $i] :
          ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).

thf(2891,axiom,
    ! [A: $i] :
    ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ARCHu_SIMPLE) ).

thf(20203,plain,
    ! [A: $i] :
    ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2891]) ).

thf(2013,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ? [D: $i] :
          ( ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDECOMPOSITION) ).

thf(15509,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ? [D: $i] :
            ( ( ( s @ ( fun @ A @ bool ) @ B )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( ( s @ ( fun @ A @ bool ) @ B )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2013]) ).

thf(2699,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_INVu_WELLDEF) ).

thf(19120,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2699]) ).

thf(2482,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALLu_MEM) ).

thf(17962,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2482]) ).

thf(1092,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ n20 @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ n20 @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_2) ).

thf(9529,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ n20 @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ n20 @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1092]) ).

thf(253,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_NEGTOTAL) ).

thf(4675,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(1273,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_CLAUSESu_LEFT) ).

thf(10650,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1273]) ).

thf(2846,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_LEu_ANTISYM) ).

thf(19937,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2846]) ).

thf(2674,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSPu_MATRIXu_NEG) ).

thf(18982,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2674]) ).

thf(647,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_SUB) ).

thf(7047,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[647]) ).

thf(449,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_absu_th) ).

thf(5942,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).

thf(2214,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
        ? [E: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
          = ( s @ B @ D ) )
    <=> ! [D: $i] :
          ( ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) ) )
        <=> ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJECTIVEu_FORALLu_THM) ).

thf(16545,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
          ? [E: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
            = ( s @ B @ D ) )
       => ! [D: $i] :
            ( ( ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) ) )
             => ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) ) )
            & ( ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) )
             => ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) ) )
             => ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) ) )
            & ( ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) )
             => ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
       => ! [D: $i] :
          ? [E: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
            = ( s @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2214]) ).

thf(2352,axiom,
    ! [A: $i] :
      ( ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ A ) ) )
     != ( s @ ind0 @ indu_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINDu_SUCu_0) ).

thf(17181,plain,
    ! [A: $i] :
      ( ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ A ) ) )
     != ( s @ ind0 @ indu_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2352]) ).

thf(924,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ D ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_BETWEEN2) ).

thf(8674,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ D ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[924]) ).

thf(482,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) )
          = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_LE) ).

thf(6098,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) )
          = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).

thf(2886,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
        ? [E: $i] :
          ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) )
          = ( s @ ( fun @ B @ bool ) @ D ) )
    <=> ! [D: $i] :
        ? [E: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
          = ( s @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJECTIVEu_IMAGE) ).

thf(20181,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
          ? [E: $i] :
            ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) )
            = ( s @ ( fun @ B @ bool ) @ D ) )
       => ! [D: $i] :
          ? [E: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
            = ( s @ B @ D ) ) )
      & ( ! [D: $i] :
          ? [E: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
            = ( s @ B @ D ) )
       => ! [D: $i] :
          ? [E: $i] :
            ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) )
            = ( s @ ( fun @ B @ bool ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2886]) ).

thf(2077,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_INTERS) ).

thf(15744,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2077]) ).

thf(1261,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ C @ bool ) @ univ ) ) ) ) ) )
      & ! [F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ C ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ C ) @ F ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ C ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ C ) @ F ) ) ) @ ( s @ num @ E ) ) ) )
      & ! [F: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ C ) @ F ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ C ) @ F ) ) ) @ ( s @ num @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_INDEXu_INRANGEu_2) ).

thf(10570,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ C @ bool ) @ univ ) ) ) ) ) )
      & ! [F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ C ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ C ) @ F ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ C ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ C ) @ F ) ) ) @ ( s @ num @ E ) ) ) )
      & ! [F: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ C ) @ F ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ C ) @ F ) ) ) @ ( s @ num @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1261]) ).

thf(191,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
        & ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ D ) @ ( s @ B @ H ) ) ) ) ) ) )
        & ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ? [I: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                  = ( s @ A @ H ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ D ) @ ( s @ B @ I ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ real ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_LEu_INCLUDED) ).

thf(4265,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
        & ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ D ) @ ( s @ B @ H ) ) ) ) ) ) )
        & ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ? [I: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                  = ( s @ A @ H ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ D ) @ ( s @ B @ I ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ real ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(995,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MINu_ACIu_conjunct4) ).

thf(9039,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[995]) ).

thf(2420,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_ACIu_conjunct0) ).

thf(17603,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2420]) ).

thf(602,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_EQu_0u_NUMSEG) ).

thf(6828,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).

thf(213,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
          & ( ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ C ) ) )
            = ( s @ ( finite_image @ A ) @ B ) ) )
      & ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
            & ( ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ C ) ) )
              = ( s @ ( finite_image @ A ) @ B ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
            & ( ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ D ) ) )
              = ( s @ ( finite_image @ A ) @ B ) ) )
         => ( ( s @ num @ C )
            = ( s @ num @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_INDEXu_WORKS) ).

thf(4383,plain,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
          & ( ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ C ) ) )
            = ( s @ ( finite_image @ A ) @ B ) ) )
      & ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
            & ( ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ C ) ) )
              = ( s @ ( finite_image @ A ) @ B ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
            & ( ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ D ) ) )
              = ( s @ ( finite_image @ A ) @ B ) ) )
         => ( ( s @ num @ C )
            = ( s @ num @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(2907,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_ASSOC) ).

thf(20259,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2907]) ).

thf(1958,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_RSUB) ).

thf(15167,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1958]) ).

thf(67,axiom,
    ! [A: $i] :
      ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ B ) ) ) )
    <=> ! [B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_VECTORu_3) ).

thf(3557,plain,
    ! [A: $i] :
      ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ B ) ) ) )
       => ! [B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ! [B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) )
       => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(376,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_UNIONu_NONZERO) ).

thf(5493,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).

thf(1963,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUCu_INJ) ).

thf(15193,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1963]) ).

thf(587,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) )
            = ( s @ ( cart @ real @ A ) @ D ) )
        <=> ! [E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
             => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBASISu_EXPANSIONu_UNIQUE) ).

thf(6734,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ D ) )
           => ! [E: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
               => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
          & ( ! [E: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
               => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) )
           => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).

thf(1669,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_MULu_EQu_conjunct0) ).

thf(13350,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1669]) ).

thf(848,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LTu_conjunct7) ).

thf(8252,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[848]) ).

thf(2315,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ E ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_COMPOSE) ).

thf(17004,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ E ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2315]) ).

thf(134,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ B @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( sum0 @ B @ A ) @ ( fun @ ( fun @ ( sum0 @ B @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( sum0 @ B @ A ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ B @ A ) ) @ inl ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ B @ A ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_CARDu_ADDu_conjunct0) ).

thf(3866,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ B @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( sum0 @ B @ A ) @ ( fun @ ( fun @ ( sum0 @ B @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( sum0 @ B @ A ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ B @ A ) ) @ inl ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ B @ A ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(2982,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_LADD) ).

thf(20675,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2982]) ).

thf(644,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) )
       => ( ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
            = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) )
          | ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
            = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) )
          | ? [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
              & ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
                = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) ) )
              & ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
                = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORDINALu_CHAINEDu_LEMMA) ).

thf(7032,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) )
       => ( ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
            = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) )
          | ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
            = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) )
          | ? [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
              & ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
                = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) ) )
              & ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
                = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[644]) ).

thf(909,axiom,
    ! [A: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ x )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_NEGu_EQu_0) ).

thf(8558,plain,
    ! [A: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ x )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ x )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[909]) ).

thf(3002,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_SYM) ).

thf(20790,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3002]) ).

thf(1382,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ bool ) @ intu_coprime ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
    <=> ? [C: $i,D: $i] :
          ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_coprime) ).

thf(11495,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ bool ) @ intu_coprime ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
       => ? [C: $i,D: $i] :
            ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      & ( ? [C: $i,D: $i] :
            ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ bool ) @ intu_coprime ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1382]) ).

thf(1338,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) ) ) )
        <=> ! [I: $i,J: $i] :
            ? [K: $i] :
              ( ( ( p @ ( s @ bool @ K ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ A @ I ) ) ) )
                  | ( ( ( s @ A @ F )
                      = ( s @ A @ I ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) ) ) )
            <=> ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ F ) ) ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) )
                & ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ F ) @ ( s @ A @ G ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ F ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_LEXu_DEPENDENT) ).

thf(11139,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) ) ) )
           => ! [I: $i,J: $i] :
              ? [K: $i] :
                ( ( ( p @ ( s @ bool @ K ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ A @ I ) ) ) )
                    | ( ( ( s @ A @ F )
                        = ( s @ A @ I ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ A @ I ) ) ) )
                    | ( ( ( s @ A @ F )
                        = ( s @ A @ I ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ B @ J ) ) ) ) ) )
                 => ( p @ ( s @ bool @ K ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
          & ( ! [I: $i,J: $i] :
              ? [K: $i] :
                ( ( ( p @ ( s @ bool @ K ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ A @ I ) ) ) )
                    | ( ( ( s @ A @ F )
                        = ( s @ A @ I ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ A @ I ) ) ) )
                    | ( ( ( s @ A @ F )
                        = ( s @ A @ I ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ B @ J ) ) ) ) ) )
                 => ( p @ ( s @ bool @ K ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) ) ) )
               => ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ F ) ) ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) ) )
              & ( ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ F ) ) ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) )
                & ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ F ) @ ( s @ A @ G ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ F ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1338]) ).

thf(22,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        <=> ? [C: $i,D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ y1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_mul) ).

thf(3268,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
           => ? [C: $i,D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ y1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) )
          & ( ? [C: $i,D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ y1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(1738,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) ) ) )
        <=> ! [G: $i,H: $i] :
            ? [I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ! [J: $i] :
                    ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) )
            <=> ! [I: $i,J: $i] :
                ? [K: $i] :
                  ( ( ( p @ ( s @ bool @ K ) )
                  <=> ! [L: $i,M: $i,N: $i] :
                        ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ M ) ) )
                          = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                       => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                          = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ F ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) ) ) )
                <=> ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) ) )
             => ! [G: $i] :
                  ( ! [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ G ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) )
                    <=> ! [I: $i,J: $i] :
                        ? [K: $i] :
                          ( ( ( p @ ( s @ bool @ K ) )
                          <=> ! [L: $i,M: $i,N: $i] :
                                ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ M ) ) )
                                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                               => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                                  = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
                 => ! [H: $i] :
                      ( ! [I: $i] :
                          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ H ) @ ( s @ B @ I ) ) )
                          = ( s @ bool @ t ) )
                     => ! [I: $i] :
                          ( ! [J: $i] :
                              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) @ ( s @ ( fun @ A @ C ) @ J ) ) )
                              = ( s @ ( fun @ B @ bool ) @ H ) )
                         => ! [J: $i] :
                              ( ! [K: $i,L: $i] :
                                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ J ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) ) ) )
                                <=> ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) ) )
                             => ! [K: $i] :
                                  ( ( ? [L: $i] :
                                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ L ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ J ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ L ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ G ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ F ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) )
                                 => ? [L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRECURSIONu_CASEWISEu_PAIRWISE) ).

thf(13715,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) ) ) )
           => ! [G: $i,H: $i] :
              ? [I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) ) )
                & ( ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
          & ( ! [G: $i,H: $i] :
              ? [I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) ) )
                & ( ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) )
               => ! [I: $i,J: $i] :
                  ? [K: $i] :
                    ( ( ( p @ ( s @ bool @ K ) )
                     => ! [L: $i,M: $i,N: $i] :
                          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ M ) ) )
                            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                         => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                            = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                    & ( ! [L: $i,M: $i,N: $i] :
                          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ M ) ) )
                            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                         => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                            = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) )
                     => ( p @ ( s @ bool @ K ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
              & ( ! [I: $i,J: $i] :
                  ? [K: $i] :
                    ( ( ( p @ ( s @ bool @ K ) )
                     => ! [L: $i,M: $i,N: $i] :
                          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ M ) ) )
                            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                         => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                            = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                    & ( ! [L: $i,M: $i,N: $i] :
                          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ M ) ) )
                            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                         => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                            = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) )
                     => ( p @ ( s @ bool @ K ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ F ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) ) ) )
                   => ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) ) )
                  & ( ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ F ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) ) ) ) ) )
             => ! [G: $i] :
                  ( ! [H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ G ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) )
                       => ! [I: $i,J: $i] :
                          ? [K: $i] :
                            ( ( ( p @ ( s @ bool @ K ) )
                             => ! [L: $i,M: $i,N: $i] :
                                  ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ M ) ) )
                                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                                 => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                            & ( ! [L: $i,M: $i,N: $i] :
                                  ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ M ) ) )
                                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                                 => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) )
                             => ( p @ ( s @ bool @ K ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
                      & ( ! [I: $i,J: $i] :
                          ? [K: $i] :
                            ( ( ( p @ ( s @ bool @ K ) )
                             => ! [L: $i,M: $i,N: $i] :
                                  ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ M ) ) )
                                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                                 => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                            & ( ! [L: $i,M: $i,N: $i] :
                                  ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ M ) ) )
                                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                                 => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) )
                             => ( p @ ( s @ bool @ K ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ G ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) ) ) )
                 => ! [H: $i] :
                      ( ! [I: $i] :
                          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ H ) @ ( s @ B @ I ) ) )
                          = ( s @ bool @ t ) )
                     => ! [I: $i] :
                          ( ! [J: $i] :
                              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) @ ( s @ ( fun @ A @ C ) @ J ) ) )
                              = ( s @ ( fun @ B @ bool ) @ H ) )
                         => ! [J: $i] :
                              ( ! [K: $i,L: $i] :
                                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ J ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) ) ) )
                                   => ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) ) )
                                  & ( ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) )
                                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ J ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) ) ) ) ) )
                             => ! [K: $i] :
                                  ( ( ? [L: $i] :
                                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ L ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ J ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ L ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ G ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ F ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) )
                                 => ? [L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1738]) ).

thf(729,axiom,
    ( ( ( ( s @ real @ y1 )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( s @ real @ y2 )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
   => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y2 ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y2 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y2 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRATu_LEMMA1) ).

thf(7500,plain,
    ( ( ( ( s @ real @ y1 )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( s @ real @ y2 )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
   => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y2 ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y2 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y2 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[729]) ).

thf(535,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_conjunct0) ).

thf(6382,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).

thf(1037,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ num ) @ neutral ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNEUTRALu_MUL) ).

thf(9266,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ num ) @ neutral ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1037]) ).

thf(2418,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_LMUL) ).

thf(17591,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2418]) ).

thf(1739,axiom,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ y ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ z ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ A @ z ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_3u_conjunct2) ).

thf(13788,plain,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ y ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ z ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ A @ z ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1739]) ).

thf(762,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EVENu_conjunct2) ).

thf(7733,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[762]) ).

thf(681,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_2) ).

thf(7236,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[681]) ).

thf(2556,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ C ) ) ) )
    <=> ( ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ C ) ) ) ) ) )
        & ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_SUMu_THM) ).

thf(18370,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ C ) ) ) )
       => ( ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ C ) ) ) ) ) )
          & ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ C ) ) ) ) ) ) ) )
      & ( ( ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ C ) ) ) ) ) )
          & ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ C ) ) ) ) ) ) )
       => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ p0 ) @ ( s @ ( sum0 @ A @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2556]) ).

thf(679,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_MULu_EQu_conjunct1) ).

thf(7226,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[679]) ).

thf(3082,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ B ) ) )
        & ( p @ ( s @ bool @ C ) ) )
    <=> ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ C ) ) )
        | ( ( p @ ( s @ bool @ B ) )
          & ( p @ ( s @ bool @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_ORu_DISTRIB) ).

thf(21249,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( ( p @ ( s @ bool @ A ) )
            | ( p @ ( s @ bool @ B ) ) )
          & ( p @ ( s @ bool @ C ) ) )
       => ( ( ( p @ ( s @ bool @ A ) )
            & ( p @ ( s @ bool @ C ) ) )
          | ( ( p @ ( s @ bool @ B ) )
            & ( p @ ( s @ bool @ C ) ) ) ) )
      & ( ( ( ( p @ ( s @ bool @ A ) )
            & ( p @ ( s @ bool @ C ) ) )
          | ( ( p @ ( s @ bool @ B ) )
            & ( p @ ( s @ bool @ C ) ) ) )
       => ( ( ( p @ ( s @ bool @ A ) )
            | ( p @ ( s @ bool @ B ) ) )
          & ( p @ ( s @ bool @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3082]) ).

thf(1250,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
    <=> ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( s @ real @ B )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
        | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_POSu_LE) ).

thf(10497,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
       => ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1250]) ).

thf(2375,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_RSUM) ).

thf(17302,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2375]) ).

thf(633,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LEu_conjunct0) ).

thf(6983,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[633]) ).

thf(2485,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) )
      = ( s @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONDu_ID) ).

thf(17979,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) )
      = ( s @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2485]) ).

thf(969,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ F ) ) ) )
        <=> ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ A @ D ) ) @ fst ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ h0 ) ) ) @ ( s @ A @ F ) ) )
            = ( s @ D @ x ) ) )
     => ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
          <=> ? [G: $i] :
                ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ A @ D ) ) @ fst ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ h0 ) ) ) @ ( s @ A @ G ) ) )
                = ( s @ D @ x ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( fun @ C @ ( fun @ D @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) ) ) @ cons ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ h0 ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ t0 ) ) ) ) ) @ ( s @ C @ f ) ) ) @ ( s @ D @ x ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ snd ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ h0 ) ) ) @ ( s @ C @ f ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( fun @ C @ ( fun @ D @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ t0 ) ) ) @ ( s @ C @ f ) ) ) @ ( s @ D @ x ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCASEWISEu_DEFu_conjunct1) ).

thf(8934,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ F ) ) ) )
           => ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ A @ D ) ) @ fst ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ h0 ) ) ) @ ( s @ A @ F ) ) )
              = ( s @ D @ x ) ) )
          & ( ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ A @ D ) ) @ fst ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ h0 ) ) ) @ ( s @ A @ F ) ) )
              = ( s @ D @ x ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
           => ? [G: $i] :
                ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ A @ D ) ) @ fst ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ h0 ) ) ) @ ( s @ A @ G ) ) )
                = ( s @ D @ x ) ) )
          & ( ? [G: $i] :
                ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ A @ D ) ) @ fst ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ h0 ) ) ) @ ( s @ A @ G ) ) )
                = ( s @ D @ x ) )
           => ( p @ ( s @ bool @ F ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( fun @ C @ ( fun @ D @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) ) ) @ cons ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ h0 ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ t0 ) ) ) ) ) @ ( s @ C @ f ) ) ) @ ( s @ D @ x ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ snd ) @ ( s @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) @ h0 ) ) ) @ ( s @ C @ f ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ ( fun @ C @ ( fun @ D @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ D ) @ ( fun @ C @ ( fun @ A @ B ) ) ) ) @ t0 ) ) ) @ ( s @ C @ f ) ) ) @ ( s @ D @ x ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).

thf(1497,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_FINITEu_SUBSETS) ).

thf(12236,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1497]) ).

thf(2427,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_REDUNDANT) ).

thf(17652,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2427]) ).

thf(2627,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_SUB) ).

thf(18773,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2627]) ).

thf(1108,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPu_FINITE) ).

thf(9636,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1108]) ).

thf(2965,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_SUBu_ABS) ).

thf(20579,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2965]) ).

thf(1951,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_UNION) ).

thf(15105,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1951]) ).

thf(478,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ y ) ) ) @ ( s @ int @ z ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ y ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ z ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAXu_ACIu_conjunct2) ).

thf(6081,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ y ) ) ) @ ( s @ int @ z ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ y ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ z ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).

thf(1671,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ADDu_LID) ).

thf(13356,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1671]) ).

thf(2092,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MINu_ASSOC) ).

thf(15878,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2092]) ).

thf(1135,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) )
    <=> ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) )
        = ( s @ int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EQu_SUBu_LADD) ).

thf(9854,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) )
          = ( s @ int @ B ) ) )
      & ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) )
          = ( s @ int @ B ) )
       => ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1135]) ).

thf(296,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ C )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_MONOu_LT) ).

thf(4988,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ C )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(1023,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_SUCu_LT) ).

thf(9204,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1023]) ).

thf(868,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ A ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_LREC) ).

thf(8394,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ A ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).

thf(346,axiom,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n10 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n10 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ A @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_1) ).

thf(5264,plain,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n10 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n10 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ A @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).

thf(1286,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_EQu_1) ).

thf(10729,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1286]) ).

thf(3148,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) )
    <=> ? [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
          & ( ( s @ ( fun @ A @ bool ) @ E )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_SUBSETu_IMAGE) ).

thf(21582,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) )
       => ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ ( fun @ A @ bool ) @ E )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
      & ( ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ ( fun @ A @ bool ) @ E )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3148]) ).

thf(2563,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NOTu_LE) ).

thf(18407,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2563]) ).

thf(3025,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) )
        <=> ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ E ) ) )
             != ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) )
              = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIRWISEu_IMAGE) ).

thf(20890,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) )
           => ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ E ) ) )
               != ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) ) ) )
          & ( ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ E ) ) )
               != ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) )
              = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3025]) ).

thf(1104,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_mulu_th) ).

thf(9593,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1104]) ).

thf(2144,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ B ) ) ) ) )
     => ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ B ) ) ) )
       => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOu_FORALL) ).

thf(16151,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ B ) ) ) ) )
     => ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ B ) ) ) )
       => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2144]) ).

thf(1468,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ card ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_NUMSEGu_LEMMA) ).

thf(12018,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ card ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1468]) ).

thf(2794,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ A @ B )
        = ( s @ A @ C ) )
    <=> ( ( s @ A @ C )
        = ( s @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_SYMu_EQ) ).

thf(19643,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ A @ B )
          = ( s @ A @ C ) )
       => ( ( s @ A @ C )
          = ( s @ A @ B ) ) )
      & ( ( ( s @ A @ C )
          = ( s @ A @ B ) )
       => ( ( s @ A @ B )
          = ( s @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2794]) ).

thf(605,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ b0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ b0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_CONVEXu_BOUND2u_LT) ).

thf(6840,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ b0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ b0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).

thf(1039,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i,C: $i] :
        ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LBOUND) ).

thf(9283,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i,C: $i] :
        ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1039]) ).

thf(1419,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_ADD) ).

thf(11778,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1419]) ).

thf(202,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUBu_conjunct2) ).

thf(4329,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(2949,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C )
        = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) )
    <=> ! [E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_EQ) ).

thf(20499,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C )
          = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) )
       => ! [E: $i] :
            ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
      & ( ! [E: $i] :
            ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
       => ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C )
          = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2949]) ).

thf(1332,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_ZERO) ).

thf(11064,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1332]) ).

thf(1975,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_CLAUSESu_conjunct6) ).

thf(15274,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1975]) ).

thf(284,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_1u_LE) ).

thf(4903,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(2293,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ measure0 ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_MEASURE) ).

thf(16929,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ measure0 ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2293]) ).

thf(1894,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ G ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) @ ( s @ ( cart @ real @ C ) @ H ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) @ ( s @ ( cart @ real @ C ) @ H ) ) ) )
         => ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) ) ) )
            <=> ( ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) ) ) )
                & ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',abilinear) ).

thf(14697,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ G ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) @ ( s @ ( cart @ real @ C ) @ H ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) @ ( s @ ( cart @ real @ C ) @ H ) ) ) )
         => ! [F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) ) ) )
               => ( ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) ) ) )
                  & ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ) )
              & ( ( ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) ) ) )
                  & ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1894]) ).

thf(2744,axiom,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_REFL) ).

thf(19338,plain,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2744]) ).

thf(167,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_LINV) ).

thf(4018,plain,
    ! [A: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(2914,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        & ( p @ ( s @ bool @ C ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_ANDu_EXISTSu_THM) ).

thf(20285,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ C ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ C ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ C ) ) )
       => ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2914]) ).

thf(2507,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
    <=> ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
           => ( ( s @ A @ C )
              = ( s @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aposet) ).

thf(18075,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
       => ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
          & ! [C: $i,D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) ) ) )
      & ( ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
          & ! [C: $i,D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2507]) ).

thf(1474,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ null ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNULLu_conjunct1) ).

thf(12051,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ null ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1474]) ).

thf(3175,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_ZERO) ).

thf(21723,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3175]) ).

thf(685,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_RSQRT) ).

thf(7247,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[685]) ).

thf(1702,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ bool ) @ isu_nadd ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) )
    <=> ? [B: $i] :
        ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aisu_nadd) ).

thf(13538,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ bool ) @ isu_nadd ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) )
       => ? [B: $i] :
          ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) )
      & ( ? [B: $i] :
          ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ bool ) @ isu_nadd ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1702]) ).

thf(961,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LETu_ADD2) ).

thf(8860,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[961]) ).

thf(2509,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ F ) ) )
            = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ G ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ G ) ) ) ) )
    <=> ? [F: $i] :
          ( ( s @ ( fun @ B @ A ) @ D )
          = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFUNCTIONu_FACTORSu_LEFT) ).

thf(18111,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i,G: $i] :
            ( ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ F ) ) )
              = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ G ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ G ) ) ) ) )
       => ? [F: $i] :
            ( ( s @ ( fun @ B @ A ) @ D )
            = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) ) )
      & ( ? [F: $i] :
            ( ( s @ ( fun @ B @ A ) @ D )
            = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) )
       => ! [F: $i,G: $i] :
            ( ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ F ) ) )
              = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ G ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2509]) ).

thf(1213,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ADDu_conjunct3) ).

thf(10271,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1213]) ).

thf(1790,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_LEu_TRANS) ).

thf(14132,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1790]) ).

thf(943,axiom,
    ( ! [A: $i] :
        ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
  <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_1) ).

thf(8802,plain,
    ( ( ! [A: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
    & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [A: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[943]) ).

thf(558,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_POW) ).

thf(6523,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).

thf(1860,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) )
        = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) )
    <=> ( ( ( s @ A @ C )
          = ( s @ A @ E ) )
        & ( ( s @ B @ D )
          = ( s @ B @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIRu_EQ) ).

thf(14532,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) )
          = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) )
       => ( ( ( s @ A @ C )
            = ( s @ A @ E ) )
          & ( ( s @ B @ D )
            = ( s @ B @ F ) ) ) )
      & ( ( ( ( s @ A @ C )
            = ( s @ A @ E ) )
          & ( ( s @ B @ D )
            = ( s @ B @ F ) ) )
       => ( ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) )
          = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1860]) ).

thf(1838,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) )
    <=> ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_EQ) ).

thf(14423,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) )
      & ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1838]) ).

thf(3040,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_SUBu_LDISTRIB) ).

thf(20997,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3040]) ).

thf(2716,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_CLAUSESu_conjunct1) ).

thf(19201,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2716]) ).

thf(2196,axiom,
    ! [A: $i,B: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_INu_EMPTY) ).

thf(16463,plain,
    ! [A: $i,B: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2196]) ).

thf(939,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_0) ).

thf(8791,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[939]) ).

thf(967,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        = ( s @ real @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_POWu_2) ).

thf(8929,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        = ( s @ real @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[967]) ).

thf(2784,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADD2u_SUB2) ).

thf(19598,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2784]) ).

thf(2471,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTEu_CASES) ).

thf(17916,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2471]) ).

thf(1862,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ A ) )
        & ( p @ ( s @ bool @ B ) ) )
    <=> ( ( p @ ( s @ bool @ B ) )
        & ( p @ ( s @ bool @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONJu_SYM) ).

thf(14547,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ B ) ) )
       => ( ( p @ ( s @ bool @ B ) )
          & ( p @ ( s @ bool @ A ) ) ) )
      & ( ( ( p @ ( s @ bool @ B ) )
          & ( p @ ( s @ bool @ A ) ) )
       => ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1862]) ).

thf(3130,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_VMUL) ).

thf(21510,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3130]) ).

thf(64,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
      <=> ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_SUPu_FINITE) ).

thf(3525,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
         => ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
        & ( ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(46,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ num ) @ neutral ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNEUTRALu_ADD) ).

thf(3438,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ num ) @ neutral ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(496,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_RNEG) ).

thf(6187,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).

thf(3098,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
        = ( s @ ( fun @ A @ bool ) @ empty ) )
    <=> ( ( s @ ( fun @ B @ bool ) @ D )
        = ( s @ ( fun @ B @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_EQu_EMPTY) ).

thf(21352,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
       => ( ( s @ ( fun @ B @ bool ) @ D )
          = ( s @ ( fun @ B @ bool ) @ empty ) ) )
      & ( ( ( s @ ( fun @ B @ bool ) @ D )
          = ( s @ ( fun @ B @ bool ) @ empty ) )
       => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3098]) ).

thf(1344,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_addu_th) ).

thf(11215,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1344]) ).

thf(2331,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
    <=> ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_SUBu_LADD) ).

thf(17090,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
          = ( s @ real @ B ) ) )
      & ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
          = ( s @ real @ B ) )
       => ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2331]) ).

thf(492,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMu_conjunct0) ).

thf(6176,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).

thf(3093,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_ADDu_SUB) ).

thf(21322,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3093]) ).

thf(1482,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) )
         => ! [D: $i,E: $i,F: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ A ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_PAIR) ).

thf(12105,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) )
         => ! [D: $i,E: $i,F: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ A ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1482]) ).

thf(2273,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJOINTu_EMPTY) ).

thf(16840,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2273]) ).

thf(80,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) )
    <=> ? [B: $i] :
        ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_FINITE) ).

thf(3621,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) )
       => ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) )
      & ( ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(160,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_LEu_CASES) ).

thf(3973,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(919,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_POW) ).

thf(8659,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[919]) ).

thf(2391,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_ADD2) ).

thf(17398,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2391]) ).

thf(1248,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LTu_conjunct3) ).

thf(10491,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1248]) ).

thf(748,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) )
     != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_SUC) ).

thf(7677,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) )
     != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[748]) ).

thf(443,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_CARD) ).

thf(5914,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).

thf(2512,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_BOUNDSu_LT) ).

thf(18126,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2512]) ).

thf(1820,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ A ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_FINITEu_IMAGE) ).

thf(14309,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ A ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1820]) ).

thf(3112,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_REFL) ).

thf(21427,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3112]) ).

thf(3163,axiom,
    ! [A: $i,B: $i] :
      ( ~ ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
    <=> ! [C: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_EXISTSu_THM) ).

thf(21656,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ! [C: $i] :
            ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
      & ( ! [C: $i] :
            ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ~ ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3163]) ).

thf(2016,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFILTERu_APPEND) ).

thf(15522,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2016]) ).

thf(1407,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
    <=> ( ( ( s @ real @ A )
          = ( s @ real @ B ) )
        | ( ( s @ real @ C )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_MULu_RCANCEL) ).

thf(11709,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
       => ( ( ( s @ real @ A )
            = ( s @ real @ B ) )
          | ( ( s @ real @ C )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      & ( ( ( ( s @ real @ A )
            = ( s @ real @ B ) )
          | ( ( s @ real @ C )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1407]) ).

thf(1370,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ m ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ n ) ) ) @ ( s @ real @ p1 ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ n ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ m ) ) ) @ ( s @ real @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_ACu_conjunct2) ).

thf(11398,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ m ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ n ) ) ) @ ( s @ real @ p1 ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ n ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ m ) ) ) @ ( s @ real @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1370]) ).

thf(69,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_TRIVu_NUMSEG) ).

thf(3569,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(1773,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_INV) ).

thf(14066,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1773]) ).

thf(1113,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_POW2) ).

thf(9680,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1113]) ).

thf(1422,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_SUBu_RADD) ).

thf(11793,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1422]) ).

thf(273,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ real @ A ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_UNIQUE) ).

thf(4798,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ real @ A ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) )
        = ( s @ real @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(1375,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ C ) ) )
     != ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asumu_DISTINCT) ).

thf(11446,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ C ) ) )
     != ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1375]) ).

thf(2109,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTEu_TOTAL) ).

thf(15975,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2109]) ).

thf(846,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAXu_ACIu_conjunct4) ).

thf(8246,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[846]) ).

thf(1703,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_MOD) ).

thf(13543,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1703]) ).

thf(490,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_UNIONSu_conjunct0) ).

thf(6157,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).

thf(1459,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_MONOu_INV) ).

thf(11971,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1459]) ).

thf(1549,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
            <=> ? [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ H ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ F ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_RESTRICTu_SET) ).

thf(12582,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
               => ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
              & ( ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ F ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1549]) ).

thf(3090,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ y ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_SUB) ).

thf(21303,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ y ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3090]) ).

thf(1120,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
        <=> ? [G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_PRODUCT) ).

thf(9781,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
           => ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
          & ( ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1120]) ).

thf(1472,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_EQu_0) ).

thf(12041,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1472]) ).

thf(2087,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_UNIVu_conjunct1) ).

thf(15834,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2087]) ).

thf(2632,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ rest ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( ( s @ A @ B )
         != ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ choice0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_REST) ).

thf(18786,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ rest ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( ( s @ A @ B )
           != ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ choice0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( ( s @ A @ B )
           != ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ choice0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ rest ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2632]) ).

thf(1331,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) ) ) )
        <=> ! [E: $i,F: $i] :
            ? [G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) @ ( s @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ bool @ G ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPOSETu_RESTRICTEDu_SUBSET) ).

thf(11036,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) ) ) )
           => ! [E: $i,F: $i] :
              ? [G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) @ ( s @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ bool @ G ) ) ) ) ) )
          & ( ! [E: $i,F: $i] :
              ? [G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) @ ( s @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ bool @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1331]) ).

thf(2499,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_ANTIMONO) ).

thf(18037,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2499]) ).

thf(2286,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_MULT2) ).

thf(16897,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2286]) ).

thf(2264,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
            & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
         => ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
              & ! [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_WELL) ).

thf(16785,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
            & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
         => ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
              & ! [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2264]) ).

thf(2789,axiom,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LETu_ANTISYM) ).

thf(19617,plain,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2789]) ).

thf(2149,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NOTu_LT) ).

thf(16204,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2149]) ).

thf(569,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ num @ B )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_EQu_0) ).

thf(6591,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).

thf(1256,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_FINITEu_SUBSETS) ).

thf(10538,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1256]) ).

thf(261,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ n20 @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ n20 @ bool ) @ univ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIMINDEXu_2) ).

thf(4707,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ n20 @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ n20 @ bool ) @ univ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(2576,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( prod @ B @ C ) @ A ) ) @ uncurry ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ E ) ) ) @ ( s @ C @ F ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ E ) ) ) @ ( s @ C @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNCURRYu_DEF) ).

thf(18475,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( prod @ B @ C ) @ A ) ) @ uncurry ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ E ) ) ) @ ( s @ C @ F ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ E ) ) ) @ ( s @ C @ F ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2576]) ).

thf(714,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_1) ).

thf(7413,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[714]) ).

thf(2809,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_THM) ).

thf(19717,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2809]) ).

thf(815,axiom,
    ! [A: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      | ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SGNu_CASES) ).

thf(8050,plain,
    ! [A: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      | ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[815]) ).

thf(755,axiom,
    ! [A: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_0) ).

thf(7707,plain,
    ! [A: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[755]) ).

thf(265,axiom,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_SUC) ).

thf(4718,plain,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(600,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
              & ( ( s @ ( fun @ A @ bool ) @ C )
               != ( s @ ( fun @ A @ bool ) @ D ) ) )
           => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
              = ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_UNIONS) ).

thf(6809,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
              & ( ( s @ ( fun @ A @ bool ) @ C )
               != ( s @ ( fun @ A @ bool ) @ D ) ) )
           => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
              = ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).

thf(2118,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_RMUL) ).

thf(16007,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2118]) ).

thf(3047,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ C ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_ID) ).

thf(21069,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ C ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3047]) ).

thf(1355,axiom,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ n ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ l0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aELu_conjunct1) ).

thf(11312,plain,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ n ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ l0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1355]) ).

thf(419,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
        <=> ! [F: $i,G: $i] :
            ? [H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ G ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
         => ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ A @ x ) ) ) )
            <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ x ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ x ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFLu_RESTRICT) ).

thf(5773,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
           => ! [F: $i,G: $i] :
              ? [H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
          & ( ! [F: $i,G: $i] :
              ? [H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
         => ! [D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ A @ x ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ x ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ x ) ) ) ) ) )
              & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ x ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ x ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ A @ x ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).

thf(2511,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( ( s @ real @ B )
        = ( s @ real @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_EQu_ODD) ).

thf(18123,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( ( s @ real @ B )
        = ( s @ real @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2511]) ).

thf(1359,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_SUBu_RADD) ).

thf(11325,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1359]) ).

thf(1924,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MINu_MIN) ).

thf(14987,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1924]) ).

thf(3031,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ C @ bool ) @ univ ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) @ ( s @ ( fun @ D @ bool ) @ univ ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ bool ) @ univ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_FUNSPACEu_CONG) ).

thf(20934,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ C @ bool ) @ univ ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) @ ( s @ ( fun @ D @ bool ) @ univ ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ bool ) @ univ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3031]) ).

thf(1192,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ! [B: $i] :
          ( ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ B ) ) ) ) )
          = ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ num @ num ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_INV) ).

thf(10145,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ! [B: $i] :
          ( ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ B ) ) ) ) )
          = ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ num @ num ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1192]) ).

thf(173,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                    = ( s @ A @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ real ) @ G ) ) ) @ ( s @ A @ H ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
             => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_IMAGEu_GEN) ).

thf(4033,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ real ) @ G ) ) ) @ ( s @ A @ H ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
             => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(796,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( list @ A ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( list @ B ) @ nil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_conjunct0) ).

thf(7931,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( list @ A ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( list @ B ) @ nil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[796]) ).

thf(1121,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ D ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ f ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ f ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ f ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_UNIONu_EQ) ).

thf(9796,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ D ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ f ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ f ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ f ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1121]) ).

thf(851,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_NEGu_MINUS1) ).

thf(8261,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[851]) ).

thf(933,axiom,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_CAUCHY) ).

thf(8774,plain,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[933]) ).

thf(2908,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ A )
        = ( s @ num @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_OFu_NUMu_WELLDEF) ).

thf(20262,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ A )
        = ( s @ num @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2908]) ).

thf(2977,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_REFL) ).

thf(20619,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2977]) ).

thf(1066,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_BOUNDu_GEN) ).

thf(9369,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1066]) ).

thf(2786,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ A @ G ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_OPu_GEN) ).

thf(19608,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ A @ G ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2786]) ).

thf(402,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_RZERO) ).

thf(5643,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).

thf(483,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) ) @ D ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ ( cart @ B @ C ) @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ C ) @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) ) ) )
     => ! [E: $i,F: $i] :
          ( ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ ( cart @ B @ C ) @ F ) ) )
          = ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ lambda ) @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) ) @ D ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ ( cart @ B @ C ) @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apastecart) ).

thf(6110,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) ) @ D ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ ( cart @ B @ C ) @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ C ) @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) ) ) )
     => ! [E: $i,F: $i] :
          ( ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ ( cart @ B @ C ) @ F ) ) )
          = ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ lambda ) @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( fun @ num @ B ) ) ) @ D ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ ( cart @ B @ C ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).

thf(360,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ u ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ u ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERSu_INSERT) ).

thf(5375,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ u ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ u ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).

thf(2708,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ D ) ) ) )
    <=> ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ D ) ) ) @ ( s @ ( cart @ B @ C ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_PASTECART) ).

thf(19177,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ D ) ) ) )
       => ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ D ) ) ) @ ( s @ ( cart @ B @ C ) @ E ) ) ) ) ) ) )
      & ( ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ D ) ) ) @ ( s @ ( cart @ B @ C ) @ E ) ) ) ) ) )
       => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ p0 ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2708]) ).

thf(96,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_powu_conjunct0) ).

thf(3685,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(861,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( sum0 @ A @ B ) @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( sum0 @ A @ B ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ F ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( sum0 @ A @ B ) @ F ) ) ) )
            <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( sum0 @ A @ B ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ G ) ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) )
              = ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ union ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aaddu_c) ).

thf(8306,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( sum0 @ A @ B ) @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( sum0 @ A @ B ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ F ) ) ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( sum0 @ A @ B ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ F ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( sum0 @ A @ B ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( sum0 @ A @ B ) @ F ) ) ) )
               => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( sum0 @ A @ B ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ G ) ) ) ) ) ) )
              & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ ( fun @ bool @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( sum0 @ A @ B ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ G ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( sum0 @ A @ B ) @ F ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) )
              = ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ union ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[861]) ).

thf(404,axiom,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n20 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n20 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ y ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ A @ y ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_2u_conjunct1) ).

thf(5648,plain,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n20 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n20 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ y ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ A @ y ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).

thf(872,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LSQRTu_LE) ).

thf(8406,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[872]) ).

thf(702,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( cart @ B @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ B ) @ g ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ g ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLAMBDAu_BETA) ).

thf(7336,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( cart @ B @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ B ) @ g ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ g ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[702]) ).

thf(1782,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ C @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ C @ B ) ) ) @ o ) @ ( s @ ( fun @ D @ B ) @ F ) ) ) @ ( s @ ( fun @ C @ D ) @ G ) ) ) ) )
      = ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ D @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ D @ B ) @ F ) ) ) ) ) @ ( s @ ( fun @ C @ D ) @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aou_ASSOC) ).

thf(14113,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ C @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ C @ B ) ) ) @ o ) @ ( s @ ( fun @ D @ B ) @ F ) ) ) @ ( s @ ( fun @ C @ D ) @ G ) ) ) ) )
      = ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ D @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ D @ B ) @ F ) ) ) ) ) @ ( s @ ( fun @ C @ D ) @ G ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1782]) ).

thf(1913,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_IMAGE) ).

thf(14874,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1913]) ).

thf(829,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_EQ) ).

thf(8141,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[829]) ).

thf(1337,axiom,
    ! [A: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_0) ).

thf(11136,plain,
    ! [A: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1337]) ).

thf(1424,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_CASES) ).

thf(11806,plain,
    ! [A: $i] :
      ( ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1424]) ).

thf(210,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ g ) @ ( s @ A @ F ) ) ) ) )
        & ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ g ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_EQu_SUPERSET) ).

thf(4364,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ g ) @ ( s @ A @ F ) ) ) ) )
        & ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ g ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(1926,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
    <=> ! [F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_FINITEu_SUBSETu_IMAGE) ).

thf(15008,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
       => ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) )
      & ( ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) )
       => ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1926]) ).

thf(299,axiom,
    ! [A: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_1u_conjunct1) ).

thf(4996,plain,
    ! [A: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(1234,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ num @ n ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) )
        = ( s @ num @ n ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIMINDEXu_UNIQUE) ).

thf(10405,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ num @ n ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) )
        = ( s @ num @ n ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1234]) ).

thf(2856,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGNEG) ).

thf(19998,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2856]) ).

thf(2561,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ F ) ) ) )
        <=> ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ F ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) ) ) )
     => ! [E: $i,F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) @ ( s @ ( prod @ B @ C ) @ F ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ ( fun @ ( prod @ B @ C ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) ) ) ) ) @ ( s @ ( prod @ B @ C ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLAMBDAu_PAIRu_THM) ).

thf(18392,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ F ) ) ) )
           => ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ F ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) ) ) )
          & ( ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ F ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ F ) ) ) ) ) )
     => ! [E: $i,F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) @ ( s @ ( prod @ B @ C ) @ F ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ ( fun @ ( prod @ B @ C ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) ) ) ) ) @ ( s @ ( prod @ B @ C ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2561]) ).

thf(93,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_MULTu_conjunct3) ).

thf(3676,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(430,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) )
            <=> ? [F: $i,G: $i] :
                  ( ( ( p @ ( s @ bool @ G ) )
                  <=> ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) )
                      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ real @ F ) ) ) ) ) )
         => ! [C: $i,D: $i] :
              ( ~ ! [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
                   => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ E ) ) )
                      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ num ) @ card ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POLYFUNu_ROOTBOUND) ).

thf(5846,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) )
               => ? [F: $i,G: $i] :
                    ( ( ( p @ ( s @ bool @ G ) )
                     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) )
                        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
                    & ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) )
                        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
                     => ( p @ ( s @ bool @ G ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ real @ F ) ) ) ) ) )
              & ( ? [F: $i,G: $i] :
                    ( ( ( p @ ( s @ bool @ G ) )
                     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) )
                        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
                    & ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) )
                        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
                     => ( p @ ( s @ bool @ G ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ real @ F ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) ) ) )
         => ! [C: $i,D: $i] :
              ( ~ ! [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
                   => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ E ) ) )
                      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ num ) @ card ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ bool ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).

thf(1630,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ a0 ) ) ) ) )
    = ( s @ int @ a0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_abstr) ).

thf(13072,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ a0 ) ) ) ) )
    = ( s @ int @ a0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1630]) ).

thf(358,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFACTu_LT) ).

thf(5343,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(927,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ measure0 ) @ ( s @ ( fun @ A @ num ) @ m ) ) ) @ ( s @ A @ B ) ) ) @ ( s @ A @ a0 ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ measure0 ) @ ( s @ ( fun @ A @ num ) @ m ) ) ) @ ( s @ A @ B ) ) ) @ ( s @ A @ b0 ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ m ) @ ( s @ A @ a0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ m ) @ ( s @ A @ b0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEASUREu_LE) ).

thf(8686,plain,
    ! [A: $i] :
      ( ( ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ measure0 ) @ ( s @ ( fun @ A @ num ) @ m ) ) ) @ ( s @ A @ B ) ) ) @ ( s @ A @ a0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ measure0 ) @ ( s @ ( fun @ A @ num ) @ m ) ) ) @ ( s @ A @ B ) ) ) @ ( s @ A @ b0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ m ) @ ( s @ A @ a0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ m ) @ ( s @ A @ b0 ) ) ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ m ) @ ( s @ A @ a0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ m ) @ ( s @ A @ b0 ) ) ) ) ) )
       => ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ measure0 ) @ ( s @ ( fun @ A @ num ) @ m ) ) ) @ ( s @ A @ B ) ) ) @ ( s @ A @ a0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ measure0 ) @ ( s @ ( fun @ A @ num ) @ m ) ) ) @ ( s @ A @ B ) ) ) @ ( s @ A @ b0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[927]) ).

thf(288,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_SUC) ).

thf(4924,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(993,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LINV) ).

thf(9035,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[993]) ).

thf(1699,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ f ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
      <=> ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ f ) @ ( s @ A @ C ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_EQu_0u_IFF) ).

thf(13521,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ f ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
         => ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
             => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ f ) @ ( s @ A @ C ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
             => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ f ) @ ( s @ A @ C ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ f ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1699]) ).

thf(1178,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_LE2u_REV) ).

thf(10057,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1178]) ).

thf(439,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_CLAUSESu_NUMSEGu_conjunct1) ).

thf(5897,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).

thf(2848,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ A ) ) ) )
    <=> ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ A ) ) ) ) )
        = ( s @ real @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_tybiju_conjunct1) ).

thf(19947,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ A ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ A ) ) ) ) )
          = ( s @ real @ A ) ) )
      & ( ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ A ) ) ) ) )
          = ( s @ real @ A ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2848]) ).

thf(2306,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_le) ).

thf(16976,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2306]) ).

thf(2859,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) ) )
     => ( ( s @ ( fun @ A @ B ) @ C )
        = ( s @ ( fun @ A @ B ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_EXT) ).

thf(20008,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) ) )
     => ( ( s @ ( fun @ A @ B ) @ C )
        = ( s @ ( fun @ A @ B ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2859]) ).

thf(2139,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_ADDu_ASSOC) ).

thf(16136,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2139]) ).

thf(2409,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPOSETu_REFL) ).

thf(17571,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2409]) ).

thf(1019,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_ADDu_SPLIT) ).

thf(9187,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1019]) ).

thf(3132,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_ANTISYM) ).

thf(21520,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3132]) ).

thf(240,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_2) ).

thf(4536,plain,
    ! [A: $i] :
      ( ( ! [B: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
       => ! [B: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(466,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_conjunct1) ).

thf(6021,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).

thf(2690,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_LNEG) ).

thf(19085,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2690]) ).

thf(1714,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_DIFF) ).

thf(13600,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1714]) ).

thf(1302,axiom,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ADDITIVE) ).

thf(10801,plain,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1302]) ).

thf(1903,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_ADD) ).

thf(14798,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1903]) ).

thf(1030,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_MULu_EQu_conjunct1) ).

thf(9236,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1030]) ).

thf(35,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ B @ real ) ) @ C ) @ ( s @ ( fun @ B @ num ) @ D ) ) ) @ ( s @ B @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ D ) @ ( s @ B @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) )
            <=> ? [I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ J ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) ) ) )
             => ! [F: $i] :
                  ( ! [G: $i,H: $i,I: $i,J: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ B @ I ) ) ) @ ( s @ A @ J ) ) ) )
                    <=> ? [K: $i,L: $i] :
                          ( ( ( p @ ( s @ bool @ L ) )
                          <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ K ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ J ) ) ) @ ( s @ bool @ L ) ) ) @ ( s @ A @ K ) ) ) ) ) )
                 => ! [G: $i,H: $i,I: $i,J: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) )
                        & ! [K: $i] :
                            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ K ) ) ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) )
                           => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ K ) ) ) ) ) ) )
                              = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ J ) @ ( s @ B @ K ) ) ) ) ) )
                     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) ) )
                        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ B @ real ) ) @ C ) @ ( s @ ( fun @ B @ num ) @ J ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_MULTICOUNTu_GEN) ).

thf(3327,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ B @ real ) ) @ C ) @ ( s @ ( fun @ B @ num ) @ D ) ) ) @ ( s @ B @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ D ) @ ( s @ B @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) )
               => ? [I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) ) )
              & ( ? [I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) ) ) )
             => ! [F: $i] :
                  ( ! [G: $i,H: $i,I: $i,J: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ B @ I ) ) ) @ ( s @ A @ J ) ) ) )
                       => ? [K: $i,L: $i] :
                            ( ( ( p @ ( s @ bool @ L ) )
                             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ K ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                            & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ K ) ) ) @ ( s @ B @ I ) ) ) ) )
                             => ( p @ ( s @ bool @ L ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ J ) ) ) @ ( s @ bool @ L ) ) ) @ ( s @ A @ K ) ) ) ) ) )
                      & ( ? [K: $i,L: $i] :
                            ( ( ( p @ ( s @ bool @ L ) )
                             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ K ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                            & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ K ) ) ) @ ( s @ B @ I ) ) ) ) )
                             => ( p @ ( s @ bool @ L ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ J ) ) ) @ ( s @ bool @ L ) ) ) @ ( s @ A @ K ) ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ B @ I ) ) ) @ ( s @ A @ J ) ) ) ) ) )
                 => ! [G: $i,H: $i,I: $i,J: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) )
                        & ! [K: $i] :
                            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ K ) ) ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) )
                           => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ K ) ) ) ) ) ) )
                              = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ J ) @ ( s @ B @ K ) ) ) ) ) )
                     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) ) )
                        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ B @ real ) ) @ C ) @ ( s @ ( fun @ B @ num ) @ J ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(2219,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
    <=> ? [F: $i] :
          ( ( ( s @ A @ C )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_IMAGE) ).

thf(16577,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
       => ? [F: $i] :
            ( ( ( s @ A @ C )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
      & ( ? [F: $i] :
            ( ( ( s @ A @ C )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2219]) ).

thf(1058,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ f ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ g ) @ ( s @ B @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ f ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ g ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_MONOu_IMAGE) ).

thf(9347,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ f ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ g ) @ ( s @ B @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ f ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ g ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1058]) ).

thf(259,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_OFu_NUMu_MUL) ).

thf(4701,plain,
    ! [A: $i,B: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(2605,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) ) @ r_r_ ) @ ( s @ real @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ r_r_ ) @ ( s @ real @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_MULu_RMUL) ).

thf(18656,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ C ) ) ) @ r_r_ ) @ ( s @ real @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ r_r_ ) @ ( s @ real @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2605]) ).

thf(15,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_POW) ).

thf(3226,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(1880,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) )
      & ? [D: $i,E: $i] :
          ( ( s @ int @ C )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ D ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_GCDu_EXISTS) ).

thf(14637,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) )
      & ? [D: $i,E: $i] :
          ( ( s @ int @ C )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ D ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1880]) ).

thf(1384,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ E ) )
      <=> ( ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ D )
            = ( s @ num @ A ) ) ) )
      & ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
          <=> ( ( s @ num @ B )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
          & ? [G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( ( s @ num @ A )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVMODu_EXISTu_0) ).

thf(11505,plain,
    ! [A: $i,B: $i] :
    ? [C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ E ) )
       => ( ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ D )
            = ( s @ num @ A ) ) ) )
      & ( ( ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ D )
            = ( s @ num @ A ) ) )
       => ( p @ ( s @ bool @ E ) ) )
      & ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
           => ( ( s @ num @ B )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
          & ( ( ( s @ num @ B )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
           => ( p @ ( s @ bool @ F ) ) )
          & ? [G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
               => ( ( ( s @ num @ A )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) )
              & ( ( ( ( s @ num @ A )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) )
               => ( p @ ( s @ bool @ G ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1384]) ).

thf(522,axiom,
    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ bool ) @ isu_nadd ) @ ( s @ ( fun @ num @ num ) @ r0 ) ) ) )
  <=> ( ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ r0 ) ) ) ) )
      = ( s @ ( fun @ num @ num ) @ r0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anaddu_rep) ).

thf(6327,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ bool ) @ isu_nadd ) @ ( s @ ( fun @ num @ num ) @ r0 ) ) ) )
     => ( ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ r0 ) ) ) ) )
        = ( s @ ( fun @ num @ num ) @ r0 ) ) )
    & ( ( ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ r0 ) ) ) ) )
        = ( s @ ( fun @ num @ num ) @ r0 ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ bool ) @ isu_nadd ) @ ( s @ ( fun @ num @ num ) @ r0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).

thf(2682,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SUB) ).

thf(19033,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2682]) ).

thf(882,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ) ) @ ( s @ int @ z ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ y ) ) ) @ ( s @ int @ z ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAXu_ACIu_conjunct1) ).

thf(8443,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) ) ) ) @ ( s @ int @ z ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ y ) ) ) @ ( s @ int @ z ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[882]) ).

thf(398,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_ASu_IMAGEu_SUBSETu_EQ) ).

thf(5629,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) ) )
      & ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).

thf(236,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        <=> ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_POWERSET) ).

thf(4519,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
          & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(2640,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
        & ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) )
         != ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aODDu_SUB) ).

thf(18814,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
          & ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) )
           != ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
          & ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) )
           != ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2640]) ).

thf(1433,axiom,
    ! [A: $i] :
      ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ B ) ) ) )
    <=> ? [B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_VECTORu_3) ).

thf(11853,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ B ) ) ) )
       => ? [B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ? [B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n3 ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1433]) ).

thf(3077,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_MONO) ).

thf(21232,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3077]) ).

thf(1647,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHypermapo_ZRu_LTu_1) ).

thf(13216,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1647]) ).

thf(3027,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ E ) ) ) @ ( s @ C @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) )
        <=> ( ( ( s @ B @ E )
              = ( s @ B @ G ) )
            & ( ( s @ C @ F )
              = ( s @ C @ H ) ) ) )
     => ? [E: $i,F: $i] :
        ! [G: $i,H: $i] :
          ( ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) )
            = ( s @ B @ G ) )
          & ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ F ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) )
            = ( s @ C @ H ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJu_INVERSE2) ).

thf(20905,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ E ) ) ) @ ( s @ C @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) )
           => ( ( ( s @ B @ E )
                = ( s @ B @ G ) )
              & ( ( s @ C @ F )
                = ( s @ C @ H ) ) ) )
          & ( ( ( ( s @ B @ E )
                = ( s @ B @ G ) )
              & ( ( s @ C @ F )
                = ( s @ C @ H ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ E ) ) ) @ ( s @ C @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) )
     => ? [E: $i,F: $i] :
        ! [G: $i,H: $i] :
          ( ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) )
            = ( s @ B @ G ) )
          & ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ F ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) )
            = ( s @ C @ H ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3027]) ).

thf(1341,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
      = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_EXPANDu_CASES) ).

thf(11195,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
      = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1341]) ).

thf(759,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_CLAUSESu_conjunct1) ).

thf(7722,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[759]) ).

thf(1146,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_MULTu_conjunct6) ).

thf(9896,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1146]) ).

thf(227,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) )
     != ( s @ ( list @ A ) @ nil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_CONSu_NIL) ).

thf(4483,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) )
     != ( s @ ( list @ A ) @ nil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(2104,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_LADD) ).

thf(15958,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2104]) ).

thf(1588,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ columns ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
          = ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',acolumns) ).

thf(12762,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ columns ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
          = ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1588]) ).

thf(410,axiom,
    ( ! [A: $i] :
        ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
  <=> ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_SUC) ).

thf(5695,plain,
    ( ( ! [A: $i] :
          ( ( ( s @ num @ A )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
     => ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ) )
    & ( ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) )
     => ! [A: $i] :
          ( ( ( s @ num @ A )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).

thf(1732,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
        & ! [G: $i,H: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
              & ( ( s @ B @ G )
               != ( s @ B @ H ) )
              & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ G ) ) )
                = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ H ) ) ) ) )
           => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ C @ ( cart @ real @ A ) ) @ D ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ G ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ C @ ( cart @ real @ A ) ) @ D ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ C @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_IMAGEu_NONZERO) ).

thf(13683,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
        & ! [G: $i,H: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
              & ( ( s @ B @ G )
               != ( s @ B @ H ) )
              & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ G ) ) )
                = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ H ) ) ) ) )
           => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ C @ ( cart @ real @ A ) ) @ D ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ G ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ C @ ( cart @ real @ A ) ) @ D ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ C @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1732]) ).

thf(2564,axiom,
    ! [A: $i] :
    ? [B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
      & ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEMMA) ).

thf(18412,plain,
    ! [A: $i] :
    ? [B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
      & ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2564]) ).

thf(1607,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( cart @ A @ B ) @ C )
        = ( s @ ( cart @ A @ B ) @ D ) )
    <=> ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARTu_EQ) ).

thf(12898,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( cart @ A @ B ) @ C )
          = ( s @ ( cart @ A @ B ) @ D ) )
       => ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
      & ( ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) )
       => ( ( s @ ( cart @ A @ B ) @ C )
          = ( s @ ( cart @ A @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1607]) ).

thf(917,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_ADDu_LDISTRIB) ).

thf(8655,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[917]) ).

thf(869,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ l0 ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aELu_TL) ).

thf(8397,plain,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ l0 ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[869]) ).

thf(42,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
      & ? [C: $i,D: $i] :
          ( ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_gcd) ).

thf(3418,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
      & ? [C: $i,D: $i] :
          ( ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(1869,axiom,
    ! [A: $i] :
      ( ( ( ( p @ ( s @ bool @ t ) )
          | ( p @ ( s @ bool @ A ) ) )
      <=> ( p @ ( s @ bool @ t ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ t ) ) )
      <=> ( p @ ( s @ bool @ t ) ) )
      & ( ( ( p @ ( s @ bool @ f0 ) )
          | ( p @ ( s @ bool @ A ) ) )
      <=> ( p @ ( s @ bool @ A ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ f0 ) ) )
      <=> ( p @ ( s @ bool @ A ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ A ) ) )
      <=> ( p @ ( s @ bool @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORu_CLAUSES) ).

thf(14569,plain,
    ! [A: $i] :
      ( ( ( ( p @ ( s @ bool @ t ) )
          | ( p @ ( s @ bool @ A ) ) )
       => ( p @ ( s @ bool @ t ) ) )
      & ( ( p @ ( s @ bool @ t ) )
       => ( ( p @ ( s @ bool @ t ) )
          | ( p @ ( s @ bool @ A ) ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ t ) ) )
       => ( p @ ( s @ bool @ t ) ) )
      & ( ( p @ ( s @ bool @ t ) )
       => ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ t ) ) ) )
      & ( ( ( p @ ( s @ bool @ f0 ) )
          | ( p @ ( s @ bool @ A ) ) )
       => ( p @ ( s @ bool @ A ) ) )
      & ( ( p @ ( s @ bool @ A ) )
       => ( ( p @ ( s @ bool @ f0 ) )
          | ( p @ ( s @ bool @ A ) ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ f0 ) ) )
       => ( p @ ( s @ bool @ A ) ) )
      & ( ( p @ ( s @ bool @ A ) )
       => ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ f0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1869]) ).

thf(1249,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_pow) ).

thf(10494,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1249]) ).

thf(672,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_ADDu_NUMSEG) ).

thf(7197,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[672]) ).

thf(3043,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
    <=> ( ( s @ ( fun @ B @ bool ) @ C )
        = ( s @ ( fun @ B @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_EMPTY) ).

thf(21009,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
       => ( ( s @ ( fun @ B @ bool ) @ C )
          = ( s @ ( fun @ B @ bool ) @ empty ) ) )
      & ( ( ( s @ ( fun @ B @ bool ) @ C )
          = ( s @ ( fun @ B @ bool ) @ empty ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3043]) ).

thf(1095,axiom,
    ! [A: $i] :
      ( ? [B: $i] :
        ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) )
    <=> ? [B: $i,C: $i] :
        ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBOUNDSu_DIVIDED) ).

thf(9556,plain,
    ! [A: $i] :
      ( ( ? [B: $i] :
          ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) )
       => ? [B: $i,C: $i] :
          ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
      & ( ? [B: $i,C: $i] :
          ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ? [B: $i] :
          ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1095]) ).

thf(2996,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ! [B: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_REALu_INTERVALu_conjunct1) ).

thf(20762,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ real @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ real @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ! [B: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2996]) ).

thf(16,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h2 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) ) @ ( s @ C @ b0 ) ) )
      = ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ h2 ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) @ ( s @ C @ b0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITLIST2u_conjunct1) ).

thf(3229,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h2 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) ) @ ( s @ C @ b0 ) ) )
      = ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ h2 ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) @ ( s @ C @ b0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(3019,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ repu_prod ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( prod @ A @ B ) ) @ absu_prod ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) ) ) )
      = ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREPu_ABSu_PAIR) ).

thf(20871,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ repu_prod ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( prod @ A @ B ) ) @ absu_prod ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) ) ) )
      = ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3019]) ).

thf(1841,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_UNIONu_SUBSET) ).

thf(14445,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1841]) ).

thf(3071,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ E ) ) ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ F ) ) ) )
        <=> ? [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( fun @ bool @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ E ) @ ( s @ ( cart @ B @ A ) @ G ) ) ) @ ( s @ ( cart @ B @ C ) @ H ) ) ) ) ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ G ) ) ) @ ( s @ ( cart @ B @ C ) @ H ) ) ) ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( fun @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ ( cart @ B @ C ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ E ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ E ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ ( cart @ B @ C ) @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_ELIMu_PASTECARTu_THM) ).

thf(21202,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ E ) ) ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ F ) ) ) )
           => ? [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( fun @ bool @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ E ) @ ( s @ ( cart @ B @ A ) @ G ) ) ) @ ( s @ ( cart @ B @ C ) @ H ) ) ) ) ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ G ) ) ) @ ( s @ ( cart @ B @ C ) @ H ) ) ) ) ) ) )
          & ( ? [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( fun @ bool @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ E ) @ ( s @ ( cart @ B @ A ) @ G ) ) ) @ ( s @ ( cart @ B @ C ) @ H ) ) ) ) ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ G ) ) ) @ ( s @ ( cart @ B @ C ) @ H ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ E ) ) ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ F ) ) ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( fun @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ ( cart @ B @ C ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ E ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ bool ) ) @ E ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ ( cart @ B @ C ) @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3071]) ).

thf(1314,axiom,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y1 ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y2 ) ) ) ) )
   => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y2 ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y2 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y2 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRATu_LEMMA2) ).

thf(10943,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y1 ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y2 ) ) ) ) )
   => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y2 ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y2 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y2 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1314]) ).

thf(2659,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ B )
       != ( s @ ( fun @ A @ bool ) @ empty ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ choice0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCHOICEu_DEF) ).

thf(18909,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ B )
       != ( s @ ( fun @ A @ bool ) @ empty ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ choice0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2659]) ).

thf(2755,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ C ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ D ) ) ) ) ) )
        & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ C ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ D ) ) ) ) ) ) )
     => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ C ) @ ( s @ ( sum0 @ A @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asumu_INDUCT) ).

thf(19401,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ C ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ D ) ) ) ) ) )
        & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ C ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ D ) ) ) ) ) ) )
     => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ C ) @ ( s @ ( sum0 @ A @ B ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2755]) ).

thf(1114,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ A @ I ) ) ) )
                <=> ? [J: $i,K: $i] :
                      ( ( ( p @ ( s @ bool @ K ) )
                      <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ I ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ A @ J ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                    & ! [J: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                       => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
                          = ( s @ num @ I ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_MULTICOUNT) ).

thf(9682,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ A @ I ) ) ) )
                   => ? [J: $i,K: $i] :
                        ( ( ( p @ ( s @ bool @ K ) )
                         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                        & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ H ) ) ) ) )
                         => ( p @ ( s @ bool @ K ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ I ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ A @ J ) ) ) ) ) )
                  & ( ? [J: $i,K: $i] :
                        ( ( ( p @ ( s @ bool @ K ) )
                         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                        & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ H ) ) ) ) )
                         => ( p @ ( s @ bool @ K ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ I ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ A @ J ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ A @ I ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                    & ! [J: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                       => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
                          = ( s @ num @ I ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ real ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1114]) ).

thf(434,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
        <=> ? [G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ G ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_PRODUCT) ).

thf(5875,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
           => ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
          & ( ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).

thf(1306,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_0) ).

thf(10811,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1306]) ).

thf(2971,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
    <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LT) ).

thf(20602,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) )
      & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2971]) ).

thf(1678,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUBu_conjunct5) ).

thf(13386,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1678]) ).

thf(1465,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_BETWEEN) ).

thf(11993,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1465]) ).

thf(282,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_REFL) ).

thf(4893,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(74,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( s @ int @ B )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ENTIRE) ).

thf(3591,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ int @ B )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      & ( ( ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ int @ B )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(547,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_POW) ).

thf(6482,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).

thf(966,axiom,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aODDu_conjunct0) ).

thf(8927,plain,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[966]) ).

thf(2954,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ g_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ageu_c) ).

thf(20528,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ g_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2954]) ).

thf(1369,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) ) @ fcons ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFCONSu_conjunct0) ).

thf(11395,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) ) @ fcons ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1369]) ).

thf(280,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ y ) ) ) @ ( s @ int @ z ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ y ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ z ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MINu_ACIu_conjunct2) ).

thf(4888,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ y ) ) ) @ ( s @ int @ z ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ y ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ z ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(844,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_ADDR) ).

thf(8233,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[844]) ).

thf(2743,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_LADD) ).

thf(19334,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2743]) ).

thf(1624,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
    <=> ? [C: $i] :
        ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anaddu_le) ).

thf(13019,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
       => ? [C: $i] :
          ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
      & ( ? [C: $i] :
          ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1624]) ).

thf(2894,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ C ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ C ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ C ) @ B ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_TRANSPu_MUL) ).

thf(20216,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ C ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ C ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ C ) @ B ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2894]) ).

thf(760,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ B ) @ A ) @ E ) @ ( s @ ( list @ B ) @ nil ) ) )
        = ( s @ A @ C ) )
      & ! [F: $i,G: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ B ) @ A ) @ E ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ F ) ) ) @ ( s @ ( list @ B ) @ G ) ) ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( fun @ A @ A ) ) ) @ D ) @ ( s @ B @ F ) ) ) @ ( s @ ( list @ B ) @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ B ) @ A ) @ E ) @ ( s @ ( list @ B ) @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alistu_RECURSION) ).

thf(7725,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ B ) @ A ) @ E ) @ ( s @ ( list @ B ) @ nil ) ) )
        = ( s @ A @ C ) )
      & ! [F: $i,G: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ B ) @ A ) @ E ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ F ) ) ) @ ( s @ ( list @ B ) @ G ) ) ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( fun @ A @ A ) ) ) @ D ) @ ( s @ B @ F ) ) ) @ ( s @ ( list @ B ) @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ B ) @ A ) @ E ) @ ( s @ ( list @ B ) @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[760]) ).

thf(586,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
         => ! [E: $i] :
              ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amatrixu_neg) ).

thf(6731,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
         => ! [E: $i] :
              ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).

thf(1160,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_REFL) ).

thf(9968,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1160]) ).

thf(2905,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) @ ( s @ ( fun @ A @ C ) @ F ) ) )
          = ( s @ ( fun @ B @ C ) @ E ) )
     => ! [E: $i,F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ B @ C ) @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPERADMISSIBLEu_CONST) ).

thf(20253,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) @ ( s @ ( fun @ A @ C ) @ F ) ) )
          = ( s @ ( fun @ B @ C ) @ E ) )
     => ! [E: $i,F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ B @ C ) @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2905]) ).

thf(226,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_POWu_R1) ).

thf(4480,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(2638,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_LADD) ).

thf(18809,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2638]) ).

thf(21,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ max ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_MAX) ).

thf(3265,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ max ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(1534,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
            <=> ? [H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ I ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ H ) ) ) @ ( s @ A @ I ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ I ) ) ) ) ) )
             => ! [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) )
                 => ! [G: $i,H: $i,I: $i,J: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                     => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ B ) @ I ) ) ) @ ( s @ ( fun @ A @ B ) @ J ) ) ) ) )
                        = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ I ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ J ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_CASES) ).

thf(12457,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) ) )
          & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
               => ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
              & ( ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ H ) ) ) @ ( s @ A @ I ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ I ) ) ) ) ) )
             => ! [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) )
                 => ! [G: $i,H: $i,I: $i,J: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                     => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ B ) @ I ) ) ) @ ( s @ ( fun @ A @ B ) @ J ) ) ) ) )
                        = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ I ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ J ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1534]) ).

thf(2389,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LTEu_TRANS) ).

thf(17389,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2389]) ).

thf(2873,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_TRIANGLEu_LT) ).

thf(20121,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2873]) ).

thf(1837,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
      | ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_CASES) ).

thf(14420,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
      | ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1837]) ).

thf(2944,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ? [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) )
                    = ( s @ A @ E ) ) )
              & ! [F: $i,G: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) )
                      = ( s @ A @ E ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ G ) ) )
                      = ( s @ A @ E ) ) )
                 => ( ( s @ A @ F )
                    = ( s @ A @ G ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ real ) ) ) @ o ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_BIJECTION) ).

thf(20480,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ? [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) )
                    = ( s @ A @ E ) ) )
              & ! [F: $i,G: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) )
                      = ( s @ A @ E ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ G ) ) )
                      = ( s @ A @ E ) ) )
                 => ( ( s @ A @ F )
                    = ( s @ A @ G ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ real ) ) ) @ o ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2944]) ).

thf(1190,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_SUBu_LZERO) ).

thf(10139,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1190]) ).

thf(1736,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i,F: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
         => ! [C: $i] :
              ( ! [D: $i,E: $i,F: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
             => ! [D: $i,E: $i,F: $i,G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ G ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ E ) ) ) @ ( s @ real @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ B ) @ ( s @ num @ G ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_POLYFUN) ).

thf(13709,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i,F: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
         => ! [C: $i] :
              ( ! [D: $i,E: $i,F: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
             => ! [D: $i,E: $i,F: $i,G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ G ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ E ) ) ) @ ( s @ real @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ B ) @ ( s @ num @ G ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1736]) ).

thf(3106,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_LSUB) ).

thf(21399,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3106]) ).

thf(3075,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ G ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ diff ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_DIFFu_INJ) ).

thf(21223,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ G ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ diff ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3075]) ).

thf(2833,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_ASSOC) ).

thf(19874,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2833]) ).

thf(2428,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ G ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ inter ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_INTERu_INJ) ).

thf(17655,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ G ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ inter ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2428]) ).

thf(1471,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
        <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ A @ G ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ! [F: $i,G: $i] :
                ( ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) )
               => ( ( s @ A @ F )
                  = ( s @ A @ G ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_IMAGEu_INJ) ).

thf(12029,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
           => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ A @ G ) ) ) ) )
          & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ A @ G ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ! [F: $i,G: $i] :
                ( ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) )
               => ( ( s @ A @ F )
                  = ( s @ A @ G ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1471]) ).

thf(2581,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_DIV) ).

thf(18498,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2581]) ).

thf(2078,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
        <=> ? [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) )
     => ! [E: $i,F: $i] :
          ( ? [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ G ) ) ) ) )
        <=> ? [G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_INu_GSPECu_conjunct1) ).

thf(15751,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
           => ? [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) )
          & ( ? [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
     => ! [E: $i,F: $i] :
          ( ( ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ G ) ) ) ) )
           => ? [G: $i,H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) ) )
          & ( ? [G: $i,H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) )
           => ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2078]) ).

thf(2047,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_LADD) ).

thf(15642,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2047]) ).

thf(782,axiom,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( p @ ( s @ bool @ D ) )
      <=> ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
      & ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
          <=> ( ( ( s @ num @ B )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
             => ( ( s @ num @ C )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ? [F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ( ( s @ num @ A )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
              & ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
                = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_EXP) ).

thf(7859,plain,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( p @ ( s @ bool @ D ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
       => ( p @ ( s @ bool @ D ) ) )
      & ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
           => ( ( ( s @ num @ B )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
             => ( ( s @ num @ C )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( ( ( s @ num @ B )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
             => ( ( s @ num @ C )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
           => ( p @ ( s @ bool @ E ) ) )
          & ? [F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
               => ( ( s @ num @ A )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
              & ( ( ( s @ num @ A )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
               => ( p @ ( s @ bool @ F ) ) )
              & ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
                = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[782]) ).

thf(1991,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ E @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) )
    <=> ! [J: $i,K: $i,L: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ F ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ L ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ F ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ E @ L ) ) ) )
            & ! [M: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ G ) @ ( s @ C @ M ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ H ) @ ( s @ E @ L ) ) ) ) ) )
               => ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ J ) @ ( s @ C @ M ) ) )
                  = ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ K ) @ ( s @ C @ M ) ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ L ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ E @ L ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aadmissible) ).

thf(15364,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ E @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) )
       => ! [J: $i,K: $i,L: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ F ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ L ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ F ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ E @ L ) ) ) )
              & ! [M: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ G ) @ ( s @ C @ M ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ H ) @ ( s @ E @ L ) ) ) ) ) )
                 => ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ J ) @ ( s @ C @ M ) ) )
                    = ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ K ) @ ( s @ C @ M ) ) ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ L ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ E @ L ) ) ) ) ) )
      & ( ! [J: $i,K: $i,L: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ F ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ L ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ F ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ E @ L ) ) ) )
              & ! [M: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ G ) @ ( s @ C @ M ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ H ) @ ( s @ E @ L ) ) ) ) ) )
                 => ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ J ) @ ( s @ C @ M ) ) )
                    = ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ K ) @ ( s @ C @ M ) ) ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ L ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ E @ L ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ E @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1991]) ).

thf(2840,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_FINITE) ).

thf(19903,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2840]) ).

thf(786,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ p1 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_ACu_conjunct2) ).

thf(7885,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ p1 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[786]) ).

thf(1463,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ g_a_u_c ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLE) ).

thf(11984,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ g_a_u_c ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1463]) ).

thf(1635,axiom,
    ! [A: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_1u_conjunct0) ).

thf(13104,plain,
    ! [A: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1635]) ).

thf(798,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_INV2) ).

thf(7937,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[798]) ).

thf(390,axiom,
    ! [A: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ hreal @ A ) ) )
      = ( s @ hreal @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_MULu_LID) ).

thf(5562,plain,
    ! [A: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ hreal @ A ) ) )
      = ( s @ hreal @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).

thf(706,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUCu_conjunct0) ).

thf(7357,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[706]) ).

thf(2362,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
         => ( ( s @ B @ E )
            = ( s @ B @ F ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_IMAGE) ).

thf(17242,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
         => ( ( s @ B @ E )
            = ( s @ B @ F ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2362]) ).

thf(2372,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_MULu_CONG) ).

thf(17289,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2372]) ).

thf(690,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_2) ).

thf(7269,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[690]) ).

thf(713,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_ADDu_SUB) ).

thf(7410,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[713]) ).

thf(1311,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ! [C: $i] :
        ? [D: $i] :
          ( ( ( s @ num @ D )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_TOTALu_LEMMA) ).

thf(10911,plain,
    ! [A: $i,B: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ! [C: $i] :
        ? [D: $i] :
          ( ( ( s @ num @ D )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1311]) ).

thf(3118,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_ASSOC) ).

thf(21444,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3118]) ).

thf(1210,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_conjunct0) ).

thf(10258,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1210]) ).

thf(1012,axiom,
    ! [A: $i] :
      ( ( ( s @ num @ u_0 )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) )
    <=> ( p @ ( s @ bool @ f0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EQu_conjunct5) ).

thf(9164,plain,
    ! [A: $i] :
      ( ( ( ( s @ num @ u_0 )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) )
       => ( p @ ( s @ bool @ f0 ) ) )
      & ( ( p @ ( s @ bool @ f0 ) )
       => ( ( s @ num @ u_0 )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1012]) ).

thf(374,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amatrixu_vectoru_mul) ).

thf(5487,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).

thf(1971,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_SETu_OFu_LIST) ).

thf(15258,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1971]) ).

thf(2739,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_SYM) ).

thf(19319,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2739]) ).

thf(955,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXu_conjunct0) ).

thf(8842,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[955]) ).

thf(1491,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_LTu_0) ).

thf(12132,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1491]) ).

thf(1269,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_LTu_1) ).

thf(10639,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1269]) ).

thf(2386,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALLu_MP) ).

thf(17363,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2386]) ).

thf(2262,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_MULu_ABSORBu_LE) ).

thf(16780,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2262]) ).

thf(2634,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_LEu_ADD) ).

thf(18796,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2634]) ).

thf(2925,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LETu_TRANS) ).

thf(20370,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2925]) ).

thf(172,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_POSu_LE) ).

thf(4031,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(1246,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_OFu_NUMu_EQ) ).

thf(10477,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1246]) ).

thf(996,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_LDIV) ).

thf(9041,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[996]) ).

thf(2021,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_LEu_WELLDEF) ).

thf(15542,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2021]) ).

thf(186,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
       => ? [B: $i] :
            ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
      & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
       => ? [B: $i] :
            ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_EXISTSu_LEMMA) ).

thf(4172,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
       => ? [B: $i] :
            ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
      & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
       => ? [B: $i] :
            ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(300,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ bool @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ bool @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_BOOL) ).

thf(4999,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ bool @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ bool @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(1140,axiom,
    ! [A: $i] :
    ? [B: $i,C: $i] :
    ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_BOUND) ).

thf(9881,plain,
    ! [A: $i] :
    ? [B: $i,C: $i] :
    ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1140]) ).

thf(1222,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_SUC) ).

thf(10345,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1222]) ).

thf(94,axiom,
    ! [A: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_MULu_RZERO) ).

thf(3679,plain,
    ! [A: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(2351,axiom,
    ( ( ( p @ ( s @ bool @ p1 ) )
      & ( p @ ( s @ bool @ q ) ) )
  <=> ( ( p @ ( s @ bool @ q ) )
      & ( p @ ( s @ bool @ p1 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONJu_ACIu_conjunct0) ).

thf(17178,plain,
    ( ( ( ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ q ) ) )
     => ( ( p @ ( s @ bool @ q ) )
        & ( p @ ( s @ bool @ p1 ) ) ) )
    & ( ( ( p @ ( s @ bool @ q ) )
        & ( p @ ( s @ bool @ p1 ) ) )
     => ( ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ q ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2351]) ).

thf(224,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ H ) ) ) )
            <=> ? [I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ J ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ I ) ) )
                        = ( s @ B @ G ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ A @ I ) ) ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                & ! [H: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_FINITEu_PREIMAGEu_GENERAL) ).

thf(4434,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ H ) ) ) )
               => ? [I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ I ) ) )
                          = ( s @ B @ G ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ I ) ) )
                          = ( s @ B @ G ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ A @ I ) ) ) ) ) )
              & ( ? [I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ I ) ) )
                          = ( s @ B @ G ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ I ) ) )
                          = ( s @ B @ G ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ A @ I ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ H ) ) ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                & ! [H: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(2868,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i] :
          ( ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) )
          = ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ u_seqpattern ) @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i] :
              ( ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) )
              = ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i,M: $i,N: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) )
                <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ J ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) )
                    & ~ ? [O: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) ) ) @ ( s @ F @ O ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i,M: $i,N: $i] :
                      ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) )
                      = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) )
                 => ! [K: $i] :
                      ( ! [L: $i,M: $i,N: $i,O: $i] :
                          ( ( s @ F @ ( i @ ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ O ) ) )
                          = ( s @ F @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ F ) @ ( i @ ( s @ ( fun @ E @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ F ) ) @ u_match ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ O ) ) ) ) ) @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ O ) ) ) ) ) )
                     => ! [L: $i] :
                          ( ! [M: $i,N: $i,O: $i] :
                              ( ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) )
                              = ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) ) )
                         => ! [M: $i] :
                              ( ! [N: $i,O: $i,P: $i,Q: $i,R: $i] :
                                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ P ) ) ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) )
                                <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) )
                                    & ? [S: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ P ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) ) ) @ ( s @ F @ S ) ) ) ) ) )
                             => ! [N: $i] :
                                  ( ! [O: $i,P: $i,Q: $i,R: $i] :
                                      ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) ) ) @ ( s @ ( fun @ B @ C ) @ R ) ) )
                                      = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) ) ) @ ( s @ ( fun @ B @ C ) @ R ) ) ) )
                                 => ! [O: $i] :
                                      ( ! [P: $i,Q: $i,R: $i,S: $i] :
                                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) ) ) @ ( s @ ( fun @ B @ C ) @ R ) ) ) @ ( s @ D @ S ) ) ) )
                                        <=> ? [T: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ B @ C ) @ R ) ) ) @ ( s @ D @ S ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) @ ( s @ ( fun @ B @ C ) @ R ) ) ) @ ( s @ D @ S ) ) ) ) ) @ ( s @ F @ T ) ) ) ) )
                                     => ! [P: $i] :
                                          ( ! [Q: $i,R: $i,S: $i] :
                                              ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ R ) ) ) @ ( s @ ( fun @ B @ C ) @ S ) ) )
                                              = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ R ) ) ) @ ( s @ ( fun @ B @ C ) @ S ) ) ) )
                                         => ! [Q: $i,R: $i,S: $i,T: $i,U: $i,V: $i] :
                                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ D @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) ) ) @ ( s @ ( fun @ D @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ T ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) ) ) @ ( s @ ( fun @ D @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ U ) ) ) ) ) ) )
                                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ D @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ U ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_MATCHu_SEQPATTERN) ).

thf(20057,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i] :
          ( ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) )
          = ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ u_seqpattern ) @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i] :
              ( ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) )
              = ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i,M: $i,N: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) )
                   => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ J ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) )
                      & ~ ? [O: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) ) ) @ ( s @ F @ O ) ) ) ) ) )
                  & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ J ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) )
                      & ~ ? [O: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) ) ) @ ( s @ F @ O ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) @ ( s @ D @ N ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i,M: $i,N: $i] :
                      ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) )
                      = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) )
                 => ! [K: $i] :
                      ( ! [L: $i,M: $i,N: $i,O: $i] :
                          ( ( s @ F @ ( i @ ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ O ) ) )
                          = ( s @ F @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ F ) @ ( i @ ( s @ ( fun @ E @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ F ) ) @ u_match ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ O ) ) ) ) ) @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ O ) ) ) ) ) )
                     => ! [L: $i] :
                          ( ! [M: $i,N: $i,O: $i] :
                              ( ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) )
                              = ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) ) )
                         => ! [M: $i] :
                              ( ! [N: $i,O: $i,P: $i,Q: $i,R: $i] :
                                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ P ) ) ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) )
                                   => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) )
                                      & ? [S: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ P ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) ) ) @ ( s @ F @ S ) ) ) ) ) )
                                  & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) )
                                      & ? [S: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ P ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) ) ) @ ( s @ F @ S ) ) ) ) )
                                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ P ) ) ) @ ( s @ ( fun @ B @ C ) @ Q ) ) ) @ ( s @ D @ R ) ) ) ) ) )
                             => ! [N: $i] :
                                  ( ! [O: $i,P: $i,Q: $i,R: $i] :
                                      ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) ) ) @ ( s @ ( fun @ B @ C ) @ R ) ) )
                                      = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) ) ) @ ( s @ ( fun @ B @ C ) @ R ) ) ) )
                                 => ! [O: $i] :
                                      ( ! [P: $i,Q: $i,R: $i,S: $i] :
                                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) ) ) @ ( s @ ( fun @ B @ C ) @ R ) ) ) @ ( s @ D @ S ) ) ) )
                                           => ? [T: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ B @ C ) @ R ) ) ) @ ( s @ D @ S ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) @ ( s @ ( fun @ B @ C ) @ R ) ) ) @ ( s @ D @ S ) ) ) ) ) @ ( s @ F @ T ) ) ) ) )
                                          & ( ? [T: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ B @ C ) @ R ) ) ) @ ( s @ D @ S ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) @ ( s @ ( fun @ B @ C ) @ R ) ) ) @ ( s @ D @ S ) ) ) ) ) @ ( s @ F @ T ) ) ) )
                                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ Q ) ) ) @ ( s @ ( fun @ B @ C ) @ R ) ) ) @ ( s @ D @ S ) ) ) ) ) )
                                     => ! [P: $i] :
                                          ( ! [Q: $i,R: $i,S: $i] :
                                              ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ R ) ) ) @ ( s @ ( fun @ B @ C ) @ S ) ) )
                                              = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ R ) ) ) @ ( s @ ( fun @ B @ C ) @ S ) ) ) )
                                         => ! [Q: $i,R: $i,S: $i,T: $i,U: $i,V: $i] :
                                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ D @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) ) ) @ ( s @ ( fun @ D @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ T ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) ) ) @ ( s @ ( fun @ D @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ U ) ) ) ) ) ) )
                                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ D @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ V ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ U ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2868]) ).

thf(1633,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_CASES) ).

thf(13094,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1633]) ).

thf(2945,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LETu_TOTAL) ).

thf(20485,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2945]) ).

thf(1946,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_DIFF) ).

thf(15082,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1946]) ).

thf(761,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_NEGTOTAL) ).

thf(7730,plain,
    ! [A: $i] :
      ( ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[761]) ).

thf(2411,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_RADD) ).

thf(17574,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2411]) ).

thf(2783,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_COUNTABLE) ).

thf(19595,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2783]) ).

thf(3129,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_ACIu_conjunct4) ).

thf(21507,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3129]) ).

thf(102,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_LE2) ).

thf(3735,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(3015,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFUNu_INu_IMAGE) ).

thf(20848,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3015]) ).

thf(2579,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_SING) ).

thf(18493,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2579]) ).

thf(263,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_UNION) ).

thf(4712,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(2186,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( ! [B: $i,C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      <=> ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSITIVEu_STEPWISEu_LTu_EQ) ).

thf(16399,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( ( ! [B: $i,C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
         => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) )
        & ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) )
         => ! [B: $i,C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2186]) ).

thf(2305,axiom,
    ( ( ( p @ ( s @ bool @ b ) )
     => ( p @ ( s @ bool @ a ) ) )
   => ( ~ ( p @ ( s @ bool @ a ) )
     => ~ ( p @ ( s @ bool @ b ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOu_NOT) ).

thf(16973,plain,
    ( ( ( p @ ( s @ bool @ b ) )
     => ( p @ ( s @ bool @ a ) ) )
   => ( ~ ( p @ ( s @ bool @ a ) )
     => ~ ( p @ ( s @ bool @ b ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2305]) ).

thf(1044,axiom,
    ( ( ( ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ q ) ) )
     => ( p @ ( s @ bool @ r0 ) ) )
  <=> ( ( p @ ( s @ bool @ q ) )
     => ( ( p @ ( s @ bool @ p1 ) )
       => ( p @ ( s @ bool @ r0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMPu_CONJu_ALT) ).

thf(9296,plain,
    ( ( ( ( ( p @ ( s @ bool @ p1 ) )
          & ( p @ ( s @ bool @ q ) ) )
       => ( p @ ( s @ bool @ r0 ) ) )
     => ( ( p @ ( s @ bool @ q ) )
       => ( ( p @ ( s @ bool @ p1 ) )
         => ( p @ ( s @ bool @ r0 ) ) ) ) )
    & ( ( ( p @ ( s @ bool @ q ) )
       => ( ( p @ ( s @ bool @ p1 ) )
         => ( p @ ( s @ bool @ r0 ) ) ) )
     => ( ( ( p @ ( s @ bool @ p1 ) )
          & ( p @ ( s @ bool @ q ) ) )
       => ( p @ ( s @ bool @ r0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1044]) ).

thf(935,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ inter ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFROMu_INTERu_NUMSEG) ).

thf(8778,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ inter ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[935]) ).

thf(3092,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) ) )
    <=> ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot ) )
        | ? [C: $i,D: $i,E: $i] :
            ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B )
              = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ E ) ) ) )
            & ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZRECSPACEu_CASES) ).

thf(21310,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) ) )
       => ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B )
            = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot ) )
          | ? [C: $i,D: $i,E: $i] :
              ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B )
                = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ E ) ) ) )
              & ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) ) ) ) )
      & ( ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B )
            = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot ) )
          | ? [C: $i,D: $i,E: $i] :
              ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B )
                = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ E ) ) ) )
              & ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3092]) ).

thf(1089,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_RINV) ).

thf(9522,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1089]) ).

thf(2249,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ univ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_LIST) ).

thf(16729,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ univ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2249]) ).

thf(2855,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
    <=> ! [B: $i] :
          ( ! [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
         => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_IND) ).

thf(19987,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
       => ! [B: $i] :
            ( ! [C: $i] :
                ( ! [D: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
           => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) )
      & ( ! [B: $i] :
            ( ! [C: $i] :
                ( ! [D: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
           => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2855]) ).

thf(1850,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_SUBSETu_conjunct1) ).

thf(14501,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1850]) ).

thf(1136,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ a0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) @ ( s @ B @ b0 ) ) )
      = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) @ ( s @ A @ a0 ) ) ) @ ( s @ B @ b0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITLISTu_EXTRA) ).

thf(9861,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ a0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) @ ( s @ B @ b0 ) ) )
      = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) @ ( s @ A @ a0 ) ) ) @ ( s @ B @ b0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1136]) ).

thf(1830,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFINITE) ).

thf(14357,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
      & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1830]) ).

thf(2736,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ E ) ) ) ) )
    <=> ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_INu_IMAGE) ).

thf(19305,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ E ) ) ) ) )
       => ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) ) ) )
      & ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) ) )
       => ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2736]) ).

thf(981,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_NEG) ).

thf(8994,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[981]) ).

thf(1417,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIRWISEu_SING) ).

thf(11768,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1417]) ).

thf(2695,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ B ) ) ) @ ( s @ A @ C ) ) ) )
    <=> ( ( s @ A @ B )
        = ( s @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aGEQu_DEF) ).

thf(19099,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ B ) ) ) @ ( s @ A @ C ) ) ) )
       => ( ( s @ A @ B )
          = ( s @ A @ C ) ) )
      & ( ( ( s @ A @ B )
          = ( s @ A @ C ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2695]) ).

thf(302,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
          <=> ( ( s @ A @ D )
              = ( s @ A @ C ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ real ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ real @ y ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ f ) @ ( s @ A @ D ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ real ) ) @ B ) @ ( s @ A @ D ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ f ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ y ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ f ) @ ( s @ A @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_CASESu_1) ).

thf(5002,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
           => ( ( s @ A @ D )
              = ( s @ A @ C ) ) )
          & ( ( ( s @ A @ D )
              = ( s @ A @ C ) )
           => ( p @ ( s @ bool @ E ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ real ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ real @ y ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ f ) @ ( s @ A @ D ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ real ) ) @ B ) @ ( s @ A @ D ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ f ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ y ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ f ) @ ( s @ A @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(1543,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
          & ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
          & ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aAPPROACHABLEu_LTu_LE) ).

thf(12510,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1543]) ).

thf(512,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_WORKS) ).

thf(6264,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).

thf(3030,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ B ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i] :
              ( ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ C ) ) ) @ ( s @ num @ D ) ) )
              = ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ! [C: $i] :
              ( ! [D: $i,E: $i,F: $i,G: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) ) @ C ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ F ) ) ) @ ( s @ num @ G ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ F ) @ ( s @ num @ G ) ) ) ) ) )
             => ! [D: $i,E: $i,F: $i,G: $i,H: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) ) @ C ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ H ) ) ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) ) @ C ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ H ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_SWAPu_NUMSEG) ).

thf(20931,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ B ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i] :
              ( ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ C ) ) ) @ ( s @ num @ D ) ) )
              = ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ! [C: $i] :
              ( ! [D: $i,E: $i,F: $i,G: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) ) @ C ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ F ) ) ) @ ( s @ num @ G ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ F ) @ ( s @ num @ G ) ) ) ) ) )
             => ! [D: $i,E: $i,F: $i,G: $i,H: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) ) @ C ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ H ) ) ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ num ) ) ) ) @ C ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ H ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3030]) ).

thf(1127,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ D ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_UNIONu_EQ) ).

thf(9817,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ D ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1127]) ).

thf(379,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i,G: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( ( s @ A @ F )
               != ( s @ A @ G ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ num ) ) ) @ o ) @ ( s @ ( fun @ B @ num ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_IMAGEu_NONZERO) ).

thf(5512,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i,G: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( ( s @ A @ F )
               != ( s @ A @ G ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ num ) ) ) @ o ) @ ( s @ ( fun @ B @ num ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(3039,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ A ) )
        | ( p @ ( s @ bool @ B ) ) )
    <=> ( ( p @ ( s @ bool @ B ) )
        | ( p @ ( s @ bool @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJu_SYM) ).

thf(20989,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ B ) ) )
       => ( ( p @ ( s @ bool @ B ) )
          | ( p @ ( s @ bool @ A ) ) ) )
      & ( ( ( p @ ( s @ bool @ B ) )
          | ( p @ ( s @ bool @ A ) ) )
       => ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3039]) ).

thf(1581,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBOUNDSu_LINEARu_0) ).

thf(12737,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1581]) ).

thf(680,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_EQu_0u_NUMSEG) ).

thf(7229,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).

thf(2652,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
         => ( ( s @ ( fun @ B @ bool ) @ E )
            = ( s @ ( fun @ B @ bool ) @ F ) ) )
    <=> ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
         => ( ( s @ B @ E )
            = ( s @ B @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJECTIVEu_ONu_IMAGE) ).

thf(18864,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
                = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
           => ( ( s @ ( fun @ B @ bool ) @ E )
              = ( s @ ( fun @ B @ bool ) @ F ) ) )
       => ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
           => ( ( s @ B @ E )
              = ( s @ B @ F ) ) ) )
      & ( ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
           => ( ( s @ B @ E )
              = ( s @ B @ F ) ) )
       => ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
                = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
           => ( ( s @ ( fun @ B @ bool ) @ E )
              = ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2652]) ).

thf(1408,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIT0u_DEFu_conjunct1) ).

thf(11719,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1408]) ).

thf(3008,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ( ( p @ ( s @ bool @ C ) )
            & ( ( s @ A @ D )
              = ( s @ A @ E ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ E ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ bool @ E ) ) ) @ ( s @ A @ D ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i] :
                  ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ A @ bool ) ) @ D ) @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) )
                  = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) ) ) )
             => ! [E: $i,F: $i] :
                  ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ A @ bool ) ) @ D ) @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) ) ) ) ) @ ( s @ A @ F ) ) )
                  = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_ELIMu_THMu_conjunct2) ).

thf(20807,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ( ( p @ ( s @ bool @ C ) )
              & ( ( s @ A @ D )
                = ( s @ A @ E ) ) ) )
          & ( ( ( p @ ( s @ bool @ C ) )
              & ( ( s @ A @ D )
                = ( s @ A @ E ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ E ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ bool @ E ) ) ) @ ( s @ A @ D ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i] :
                  ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ A @ bool ) ) @ D ) @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) )
                  = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) ) ) )
             => ! [E: $i,F: $i] :
                  ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ A @ bool ) ) @ D ) @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) ) ) ) ) @ ( s @ A @ F ) ) )
                  = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3008]) ).

thf(2852,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_ANTISYM) ).

thf(19967,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) )
       => ( ( s @ real @ A )
          = ( s @ real @ B ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ B ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2852]) ).

thf(1396,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_EQu_0) ).

thf(11587,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1396]) ).

thf(2151,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i] :
          ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_EQu_NEUTRAL) ).

thf(16215,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i] :
          ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2151]) ).

thf(1347,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ? [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ D ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ D ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ E ) ) ) ) )
             => ( ( s @ num @ D )
                = ( s @ num @ E ) ) )
          & ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_INDEXu_NUMSEG) ).

thf(11223,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ? [C: $i] :
            ( ! [D: $i,E: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ D ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ D ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ E ) ) ) ) )
               => ( ( s @ num @ D )
                  = ( s @ num @ E ) ) )
            & ( ( s @ ( fun @ A @ bool ) @ B )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ) ) )
      & ( ? [C: $i] :
            ( ! [D: $i,E: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ D ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ D ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ E ) ) ) ) )
               => ( ( s @ num @ D )
                  = ( s @ num @ E ) ) )
            & ( ( s @ ( fun @ A @ bool ) @ B )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1347]) ).

thf(2700,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ B )
       != ( s @ ( fun @ A @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMBERu_NOTu_EMPTY) ).

thf(19122,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
       => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2700]) ).

thf(454,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_ADDR) ).

thf(5956,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).

thf(2631,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_TRANS) ).

thf(18784,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2631]) ).

thf(1435,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ C )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_MONO) ).

thf(11865,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ C )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1435]) ).

thf(425,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_CLAUSESu_LEFT) ).

thf(5815,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).

thf(1804,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ODDu_conjunct0) ).

thf(14222,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1804]) ).

thf(2880,axiom,
    ! [A: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) )
        & ! [B: $i] :
            ( ! [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUP) ).

thf(20152,plain,
    ! [A: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) )
        & ! [B: $i] :
            ( ! [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2880]) ).

thf(1345,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LTu_conjunct0) ).

thf(11217,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1345]) ).

thf(2201,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
        ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
    <=> ? [C: $i,D: $i] :
        ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBOUNDSu_IGNORE) ).

thf(16484,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
          ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ? [C: $i,D: $i] :
          ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) )
      & ( ? [C: $i,D: $i] :
          ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
       => ? [C: $i] :
          ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2201]) ).

thf(756,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_LEu_0) ).

thf(7710,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[756]) ).

thf(1920,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
        = ( s @ num @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_CARD) ).

thf(14973,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
        = ( s @ num @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1920]) ).

thf(3147,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) ) @ vectoru_matrixu_mul ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_LMULu_MATRIX) ).

thf(21579,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) ) @ vectoru_matrixu_mul ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3147]) ).

thf(1276,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_LT0) ).

thf(10657,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1276]) ).

thf(1024,axiom,
    ! [A: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
      <=> ? [B: $i] :
            ( ( s @ ( fun @ real @ bool ) @ A )
            = ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ empty ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUPu_EQu_INF) ).

thf(9207,plain,
    ! [A: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
         => ? [B: $i] :
              ( ( s @ ( fun @ real @ bool ) @ A )
              = ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ empty ) ) ) ) )
        & ( ? [B: $i] :
              ( ( s @ ( fun @ real @ bool ) @ A )
              = ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ empty ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1024]) ).

thf(2432,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i,I: $i] :
          ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) )
          = ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ ( fun @ C @ C ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ F ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) @ ( s @ ( fun @ A @ C ) @ I ) ) )
              = ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i,I: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) )
                <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) )
             => ! [G: $i] :
                  ( ! [H: $i,I: $i] :
                      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ A @ C ) @ I ) ) )
                      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) )
                 => ! [H: $i] :
                      ( ! [I: $i,J: $i,K: $i,L: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ A @ C ) @ K ) ) ) @ ( s @ B @ L ) ) ) )
                        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) @ ( s @ ( fun @ A @ C ) @ K ) ) ) @ ( s @ B @ L ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ K ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                     => ! [I: $i] :
                          ( ! [J: $i,K: $i,L: $i] :
                              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ A @ C ) @ L ) ) )
                              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) )
                         => ! [J: $i,K: $i,L: $i,M: $i,N: $i,O: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ L ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ L ) ) ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ L ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ O ) ) ) ) )
                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ O ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPERADMISSIBLEu_COND) ).

thf(17683,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i,I: $i] :
          ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) )
          = ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ ( fun @ C @ C ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ F ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) @ ( s @ ( fun @ A @ C ) @ I ) ) )
              = ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) )
                   => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) )
                  & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ C ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) )
             => ! [G: $i] :
                  ( ! [H: $i,I: $i] :
                      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ A @ C ) @ I ) ) )
                      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) )
                 => ! [H: $i] :
                      ( ! [I: $i,J: $i,K: $i,L: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ A @ C ) @ K ) ) ) @ ( s @ B @ L ) ) ) )
                           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) @ ( s @ ( fun @ A @ C ) @ K ) ) ) @ ( s @ B @ L ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ K ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) @ ( s @ ( fun @ A @ C ) @ K ) ) ) @ ( s @ B @ L ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ K ) ) ) @ ( s @ B @ L ) ) ) ) )
                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ A @ C ) @ K ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                     => ! [I: $i] :
                          ( ! [J: $i,K: $i,L: $i] :
                              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ A @ C ) @ L ) ) )
                              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) )
                         => ! [J: $i,K: $i,L: $i,M: $i,N: $i,O: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ L ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ L ) ) ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ L ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ O ) ) ) ) )
                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ O ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2432]) ).

thf(2018,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ num @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ ( fun @ ( fun @ num @ bool ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ) @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ univ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_REAL) ).

thf(15531,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ num @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ ( fun @ ( fun @ num @ bool ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ) @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ univ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2018]) ).

thf(2519,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ i1 ) @ ( s @ A @ B ) ) )
      = ( s @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIu_THM) ).

thf(18151,plain,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ i1 ) @ ( s @ A @ B ) ) )
      = ( s @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2519]) ).

thf(2887,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_SUBu_RDISTRIB) ).

thf(20188,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2887]) ).

thf(907,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ invertible ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) )
     => ( ( ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_inv ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) ) )
          = ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
        & ( ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_inv ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) )
          = ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_INV) ).

thf(8550,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ invertible ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) )
     => ( ( ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_inv ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) ) )
          = ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
        & ( ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_inv ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) )
          = ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[907]) ).

thf(653,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
            <=> ? [H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ I ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) @ ( s @ A @ H ) ) )
                       != ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) ) ) ) ) ) )
             => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) ) )
                = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_ADDu_GEN) ).

thf(7084,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
               => ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) @ ( s @ A @ H ) ) )
                         != ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) @ ( s @ A @ H ) ) )
                         != ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
              & ( ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) @ ( s @ A @ H ) ) )
                         != ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) @ ( s @ A @ H ) ) )
                         != ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) ) ) ) ) ) )
             => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) ) )
                = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[653]) ).

thf(1857,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_FUNSPACEu_UNIV) ).

thf(14524,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1857]) ).

thf(1162,axiom,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ x ) ) ) @ ( s @ nadd @ y ) ) ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ x ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ y ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ahrealu_mulu_th) ).

thf(9981,plain,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ x ) ) ) @ ( s @ nadd @ y ) ) ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ x ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ y ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1162]) ).

thf(1008,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_ABSu_LE) ).

thf(9126,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1008]) ).

thf(1596,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LE2u_REV) ).

thf(12846,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1596]) ).

thf(1569,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aELu_APPEND) ).

thf(12687,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1569]) ).

thf(618,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_2) ).

thf(6915,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[618]) ).

thf(232,axiom,
    ? [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ hreal @ ( i @ ( s @ ( fun @ real @ hreal ) @ A ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ B ) @ ( s @ hreal @ C ) ) ) ) )
          = ( s @ hreal @ C ) )
      & ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ B ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ real @ hreal ) @ A ) @ ( s @ real @ C ) ) ) ) )
            = ( s @ real @ C ) ) )
      & ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ B ) @ ( s @ hreal @ C ) ) ) ) ) )
      & ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ B ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ B ) @ ( s @ hreal @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_HREALu_LEMMA2) ).

thf(4499,plain,
    ? [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ hreal @ ( i @ ( s @ ( fun @ real @ hreal ) @ A ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ B ) @ ( s @ hreal @ C ) ) ) ) )
          = ( s @ hreal @ C ) )
      & ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ B ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ real @ hreal ) @ A ) @ ( s @ real @ C ) ) ) ) )
            = ( s @ real @ C ) ) )
      & ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ B ) @ ( s @ hreal @ C ) ) ) ) ) )
      & ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ B ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ B ) @ ( s @ hreal @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(1577,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_negu_th) ).

thf(12715,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1577]) ).

thf(230,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ E ) @ ( s @ C @ F ) ) ) @ ( s @ D @ G ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ f ) @ ( s @ C @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ g ) @ ( s @ D @ G ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i] :
              ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ F ) @ ( s @ C @ G ) ) )
              = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ E ) @ ( s @ C @ G ) ) ) )
         => ! [G: $i,H: $i] :
              ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ C @ A ) @ f ) ) ) @ ( s @ ( list @ C ) @ G ) ) ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ D ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( list @ D ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ D @ B ) @ g ) ) ) @ ( s @ ( list @ D ) @ H ) ) ) ) )
              = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( fun @ ( list @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( list @ C ) @ ( fun @ ( list @ D ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ F ) ) ) @ ( s @ ( list @ C ) @ G ) ) ) @ ( s @ ( list @ D ) @ H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALL2u_MAP2) ).

thf(4493,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ E ) @ ( s @ C @ F ) ) ) @ ( s @ D @ G ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ f ) @ ( s @ C @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ g ) @ ( s @ D @ G ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i] :
              ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ F ) @ ( s @ C @ G ) ) )
              = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ E ) @ ( s @ C @ G ) ) ) )
         => ! [G: $i,H: $i] :
              ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ C @ A ) @ f ) ) ) @ ( s @ ( list @ C ) @ G ) ) ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ D ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( list @ D ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ D @ B ) @ g ) ) ) @ ( s @ ( list @ D ) @ H ) ) ) ) )
              = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( fun @ ( list @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( list @ C ) @ ( fun @ ( list @ D ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ F ) ) ) @ ( s @ ( list @ C ) @ G ) ) ) @ ( s @ ( list @ D ) @ H ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(316,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ G ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_PARTIALu_PRE) ).

thf(5085,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ G ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).

thf(2958,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_WELLDEF) ).

thf(20550,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2958]) ).

thf(125,axiom,
    ( ( ( s @ num @ u_0 )
      = ( s @ num @ u_0 ) )
  <=> ( p @ ( s @ bool @ t ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EQu_conjunct1) ).

thf(3816,plain,
    p @ ( s @ bool @ t ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(806,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
        | ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aODDu_EXP) ).

thf(7984,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
          | ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
          | ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[806]) ).

thf(1696,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) ) ) )
        <=> ! [E: $i,F: $i] :
            ? [G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) @ ( s @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ bool @ G ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) )
          = ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFLu_RESTRICTEDu_SUBSET) ).

thf(13498,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) ) ) )
           => ! [E: $i,F: $i] :
              ? [G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) @ ( s @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ bool @ G ) ) ) ) ) )
          & ( ! [E: $i,F: $i] :
              ? [G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) @ ( s @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ bool @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) )
          = ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1696]) ).

thf(438,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_NZ) ).

thf(5893,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).

thf(2464,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ( ( s @ A @ D )
            = ( s @ A @ C ) ) )
     => ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) ) )
          = ( s @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSELECTu_REFL) ).

thf(17876,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ( ( s @ A @ D )
              = ( s @ A @ C ) ) )
          & ( ( ( s @ A @ D )
              = ( s @ A @ C ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) ) )
          = ( s @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2464]) ).

thf(124,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_NUMSEG) ).

thf(3814,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(1883,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_RADD) ).

thf(14648,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1883]) ).

thf(1343,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_RID) ).

thf(11212,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1343]) ).

thf(9,axiom,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ k ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ k ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLASTu_CLAUSESu_conjunct1) ).

thf(3207,plain,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ k ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ k ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(684,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GEu_conjunct2) ).

thf(7244,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[684]) ).

thf(2168,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_MULu_ASSOC) ).

thf(16307,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2168]) ).

thf(1334,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) )
        <=> ! [E: $i,F: $i] :
            ? [G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ a0 ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) ) ) @ ( s @ bool @ G ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
         => ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ A @ D ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ a0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXTENDu_FL) ).

thf(11074,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) )
           => ! [E: $i,F: $i] :
              ? [G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ a0 ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ a0 ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) ) ) @ ( s @ bool @ G ) ) ) ) ) )
          & ( ! [E: $i,F: $i] :
              ? [G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ a0 ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ a0 ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) ) ) @ ( s @ bool @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
         => ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ A @ D ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ a0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1334]) ).

thf(95,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ C ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ C ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ C ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ C ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_MULu_RZERO) ).

thf(3682,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ C ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ C ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ C ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ C ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(168,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ m ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ( ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ m ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ahrealu_ofu_num) ).

thf(4021,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ m ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ( ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ m ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(1115,axiom,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ y ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ z ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ A @ y ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_3u_conjunct1) ).

thf(9734,plain,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n3 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n3 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n3 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ y ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ z ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ A @ y ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1115]) ).

thf(1365,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_PARTIALu_SUC) ).

thf(11371,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ E ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1365]) ).

thf(1713,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
      <=> ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
            = ( s @ num @ A ) ) ) )
      & ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
          <=> ( ( s @ num @ B )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
          & ? [E: $i] :
              ( ( ( p @ ( s @ bool @ E ) )
              <=> ( ( ( s @ num @ A )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ C ) ) ) @ ( s @ bool @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVISIONu_0) ).

thf(13583,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
       => ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
            = ( s @ num @ A ) ) ) )
      & ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
            = ( s @ num @ A ) ) )
       => ( p @ ( s @ bool @ C ) ) )
      & ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
           => ( ( s @ num @ B )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
          & ( ( ( s @ num @ B )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
           => ( p @ ( s @ bool @ D ) ) )
          & ? [E: $i] :
              ( ( ( p @ ( s @ bool @ E ) )
               => ( ( ( s @ num @ A )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) )
              & ( ( ( ( s @ num @ A )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
               => ( p @ ( s @ bool @ E ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ C ) ) ) @ ( s @ bool @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1713]) ).

thf(2953,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ A ) ) ) )
    <=> ? [B: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ainteger) ).

thf(20521,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ A ) ) ) )
       => ? [B: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      & ( ? [B: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2953]) ).

thf(137,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ B ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_CONSTu_NUMSEG) ).

thf(3889,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ B ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(2478,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) )
      = ( s @ ( fun @ A @ bool ) @ p1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_ACIu_conjunct3) ).

thf(17945,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) )
      = ( s @ ( fun @ A @ bool ) @ p1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2478]) ).

thf(3050,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( ! [J: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ J ) ) ) ) )
                = ( s @ B @ J ) ) ) )
        & ! [J: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) ) ) )
                = ( s @ A @ J ) )
              & ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ F ) @ ( s @ A @ J ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ F ) ) )
        = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_EQu_GENERALu_INVERSES) ).

thf(21083,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( ! [J: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ J ) ) ) ) )
                = ( s @ B @ J ) ) ) )
        & ! [J: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) ) ) )
                = ( s @ A @ J ) )
              & ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ F ) @ ( s @ A @ J ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ F ) ) )
        = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3050]) ).

thf(1983,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) )
    <=> ( ( s @ hreal @ A )
        = ( s @ hreal @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_EQu_ADDu_RCANCEL) ).

thf(15298,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) )
          = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) )
       => ( ( s @ hreal @ A )
          = ( s @ hreal @ B ) ) )
      & ( ( ( s @ hreal @ A )
          = ( s @ hreal @ B ) )
       => ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) )
          = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1983]) ).

thf(2036,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_ABS) ).

thf(15595,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2036]) ).

thf(1015,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_LE0) ).

thf(9177,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1015]) ).

thf(23,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) )
          = ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anaddu_mul) ).

thf(3282,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) )
          = ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(2235,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_RULESu_conjunct1) ).

thf(16652,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2235]) ).

thf(2056,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_UNIONS) ).

thf(15670,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2056]) ).

thf(945,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GTu_conjunct4) ).

thf(8810,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[945]) ).

thf(2565,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_MAX) ).

thf(18415,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2565]) ).

thf(1280,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ A @ B ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( cart @ A @ B ) @ bool ) @ univ ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_CARTu_UNIV) ).

thf(10710,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ A @ B ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( cart @ A @ B ) @ bool ) @ univ ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1280]) ).

thf(2438,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_MAX) ).

thf(17738,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2438]) ).

thf(849,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_CLAUSESu_LEFT) ).

thf(8255,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[849]) ).

thf(2357,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) )
     => ! [E: $i,F: $i,G: $i,H: $i,I: $i] :
          ( ( ! [J: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
               => ( ? [K: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ K ) ) )
                        = ( s @ B @ J ) ) )
                  & ! [K: $i,L: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ K ) ) )
                          = ( s @ B @ J ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ L ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ L ) ) )
                          = ( s @ B @ J ) ) )
                     => ( ( s @ A @ K )
                        = ( s @ A @ L ) ) ) ) )
            & ! [J: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ J ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                  & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ H ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ J ) ) ) ) )
                    = ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ J ) ) ) ) ) ) )
         => ( ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ C ) @ G ) ) )
            = ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_EQu_GENERAL) ).

thf(17200,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) )
     => ! [E: $i,F: $i,G: $i,H: $i,I: $i] :
          ( ( ! [J: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
               => ( ? [K: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ K ) ) )
                        = ( s @ B @ J ) ) )
                  & ! [K: $i,L: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ K ) ) )
                          = ( s @ B @ J ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ L ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ L ) ) )
                          = ( s @ B @ J ) ) )
                     => ( ( s @ A @ K )
                        = ( s @ A @ L ) ) ) ) )
            & ! [J: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ J ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                  & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ H ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ J ) ) ) ) )
                    = ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ J ) ) ) ) ) ) )
         => ( ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ C ) @ G ) ) )
            = ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2357]) ).

thf(1724,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ u_destu_rec ) @ ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ C ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ E ) ) )
          = ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( recspace @ A ) ) @ u_mku_rec ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ E ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONSTR) ).

thf(13654,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ u_destu_rec ) @ ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ C ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ E ) ) )
          = ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( recspace @ A ) ) @ u_mku_rec ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ E ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1724]) ).

thf(2817,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) )
    <=> ( ( s @ int @ A )
        = ( s @ int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EQu_NEG2) ).

thf(19751,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) )
       => ( ( s @ int @ A )
          = ( s @ int @ B ) ) )
      & ( ( ( s @ int @ A )
          = ( s @ int @ B ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2817]) ).

thf(411,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ C @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ? [J: $i,K: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ in ) @ ( s @ C @ H ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ C @ H ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) )
            <=> ? [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
         => ! [F: $i,G: $i] :
              ( ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( fun @ C @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) ) ) ) )
              = ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) @ gspec ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_GSPECu_conjunct1) ).

thf(5702,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ C @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ? [J: $i,K: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ in ) @ ( s @ C @ H ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) ) ) )
                & ( ? [J: $i,K: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ in ) @ ( s @ C @ H ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ C @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ? [J: $i,K: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ in ) @ ( s @ C @ H ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) ) ) )
                & ( ? [J: $i,K: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ in ) @ ( s @ C @ H ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ C @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ C @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) )
               => ? [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
              & ( ? [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) ) ) )
         => ! [F: $i,G: $i] :
              ( ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( fun @ C @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) ) ) ) )
              = ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) @ gspec ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).

thf(1391,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) )
          = ( s @ bool @ t ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) @ ( s @ ( fun @ A @ C ) @ F ) ) )
              = ( s @ ( fun @ B @ bool ) @ D ) )
         => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ t0 ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ t0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPERADMISSIBLEu_T) ).

thf(11570,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) )
          = ( s @ bool @ t ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) @ ( s @ ( fun @ A @ C ) @ F ) ) )
              = ( s @ ( fun @ B @ bool ) @ D ) )
         => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ t0 ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ t0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1391]) ).

thf(2752,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ real @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ real @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ real @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_RMUL) ).

thf(19390,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ real @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ real @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ real @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2752]) ).

thf(1313,axiom,
    ( ( ( p @ ( s @ bool @ p1 ) )
      | ( p @ ( s @ bool @ q ) )
      | ( p @ ( s @ bool @ r0 ) ) )
  <=> ( ( p @ ( s @ bool @ q ) )
      | ( p @ ( s @ bool @ p1 ) )
      | ( p @ ( s @ bool @ r0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJu_ACIu_conjunct2) ).

thf(10939,plain,
    ( ( ( ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ q ) )
        | ( p @ ( s @ bool @ r0 ) ) )
     => ( ( p @ ( s @ bool @ q ) )
        | ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ r0 ) ) ) )
    & ( ( ( p @ ( s @ bool @ q ) )
        | ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ r0 ) ) )
     => ( ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ q ) )
        | ( p @ ( s @ bool @ r0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1313]) ).

thf(2854,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_BOUNDSu_LT) ).

thf(19980,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2854]) ).

thf(407,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ODDu_conjunct3) ).

thf(5655,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).

thf(1861,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_INTER) ).

thf(14541,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1861]) ).

thf(2182,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
    <=> ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSWAPu_EXISTSu_THM) ).

thf(16371,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
       => ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ D ) ) ) ) )
      & ( ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ D ) ) ) )
       => ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2182]) ).

thf(1656,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_subu_th) ).

thf(13282,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1656]) ).

thf(811,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
      <=> ? [C: $i] :
            ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_EQu_0) ).

thf(8026,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
         => ? [C: $i] :
              ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) )
        & ( ? [C: $i] :
              ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[811]) ).

thf(929,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ! [H: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ bool ) ) @ mem ) @ ( s @ B @ H ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ bool @ ( fun @ ( list @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ B ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( list @ A ) @ F ) ) ) )
            <=> ? [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ H ) )
                  <=> ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ I ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ A ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( list @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( fun @ ( list @ B ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ B ) @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_LIST) ).

thf(8700,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ! [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ bool ) ) @ mem ) @ ( s @ B @ H ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
                & ( ! [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ bool ) ) @ mem ) @ ( s @ B @ H ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ bool @ ( fun @ ( list @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ B ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ! [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ bool ) ) @ mem ) @ ( s @ B @ H ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
                & ( ! [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ bool ) ) @ mem ) @ ( s @ B @ H ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ bool @ ( fun @ ( list @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ B ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( list @ A ) @ F ) ) ) )
               => ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ! [I: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ I ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
                    & ( ! [I: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ I ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ A ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) ) ) )
              & ( ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ! [I: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ I ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
                    & ( ! [I: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ I ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ A ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( list @ A ) @ G ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( list @ A ) @ F ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( list @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( fun @ ( list @ B ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ B ) @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[929]) ).

thf(669,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
          = ( s @ real @ B ) )
      <=> ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_LDIVu_EQ) ).

thf(7185,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
            = ( s @ real @ B ) )
         => ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        & ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
            = ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[669]) ).

thf(1001,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_LMUL) ).

thf(9058,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1001]) ).

thf(1808,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_EQ) ).

thf(14248,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1808]) ).

thf(607,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_GPu_MULTIPLIED) ).

thf(6848,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).

thf(1508,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POW2u_ABS) ).

thf(12279,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1508]) ).

thf(6,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) )
      = ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) ) @ cons ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ B ) @ B ) @ hd ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ tl ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZIPu_DEFu_conjunct1) ).

thf(3198,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) )
      = ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) ) @ cons ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ B ) @ B ) @ hd ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ tl ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(2816,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_NEGu_SUB) ).

thf(19748,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2816]) ).

thf(2599,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( prod @ hreal @ hreal ) @ A )
        = ( s @ ( prod @ hreal @ hreal ) @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_EQu_AP) ).

thf(18632,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( prod @ hreal @ hreal ) @ A )
        = ( s @ ( prod @ hreal @ hreal ) @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2599]) ).

thf(2265,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_MUL) ).

thf(16790,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2265]) ).

thf(1993,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ G ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ H ) ) ) )
            & ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ F ) @ ( s @ C @ G ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ F ) @ ( s @ C @ H ) ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ G ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ H ) ) ) ) )
    <=> ? [G: $i] :
        ! [H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ H ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ H ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ F ) @ ( s @ C @ H ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFUNCTIONu_FACTORSu_LEFTu_GEN) ).

thf(15383,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ! [G: $i,H: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ G ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ H ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ F ) @ ( s @ C @ G ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ F ) @ ( s @ C @ H ) ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ G ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ H ) ) ) ) )
       => ? [G: $i] :
          ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ H ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ H ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ F ) @ ( s @ C @ H ) ) ) ) ) ) ) )
      & ( ? [G: $i] :
          ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ H ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ H ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ F ) @ ( s @ C @ H ) ) ) ) ) ) )
       => ! [G: $i,H: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ G ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ D ) @ ( s @ C @ H ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ F ) @ ( s @ C @ G ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ F ) @ ( s @ C @ H ) ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ G ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ H ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1993]) ).

thf(1715,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_MULTu_conjunct2) ).

thf(13612,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1715]) ).

thf(2116,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_UNIONu_ABSORPTION) ).

thf(15994,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ C ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ C ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2116]) ).

thf(298,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFACTu_conjunct0) ).

thf(4994,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(1839,axiom,
    ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GTu_conjunct1) ).

thf(14430,plain,
    ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1839]) ).

thf(2535,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_BOUNDSu_LE) ).

thf(18283,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2535]) ).

thf(1103,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i,G: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
           => ( ( s @ B @ F )
              = ( s @ B @ G ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_IMAGEu_INJ) ).

thf(9586,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i,G: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
           => ( ( s @ B @ F )
              = ( s @ B @ G ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1103]) ).

thf(2421,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ univ )
     != ( s @ ( fun @ A @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIVu_NOTu_EMPTY) ).

thf(17606,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ univ )
     != ( s @ ( fun @ A @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2421]) ).

thf(72,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_LE2u_ODD) ).

thf(3578,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(2054,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_CAUCHYu_SCHWARZ) ).

thf(15665,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2054]) ).

thf(1949,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_ADDR) ).

thf(15093,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1949]) ).

thf(930,axiom,
    ! [A: $i] :
      ( ( ( s @ hreal @ A )
       != ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ A ) ) ) ) ) @ ( s @ hreal @ A ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_MULu_LINV) ).

thf(8736,plain,
    ! [A: $i] :
      ( ( ( s @ hreal @ A )
       != ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ A ) ) ) ) ) @ ( s @ hreal @ A ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[930]) ).

thf(1708,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_RID) ).

thf(13568,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1708]) ).

thf(2991,axiom,
    ? [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ind0 @ ind0 ) @ bool ) @ oneu_one ) @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) ) ) )
      & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ind0 @ ind0 ) @ bool ) @ onto ) @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFINITYu_AX) ).

thf(20721,plain,
    ? [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ind0 @ ind0 ) @ bool ) @ oneu_one ) @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) ) ) )
      & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ind0 @ ind0 ) @ bool ) @ onto ) @ ( s @ ( fun @ ind0 @ ind0 ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2991]) ).

thf(730,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) )
        <=> ? [G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ( ! [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                  & ! [I: $i] :
                      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                     => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                        = ( s @ A @ d ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ A ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_FUNSPACE) ).

thf(7503,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) )
           => ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                    & ! [I: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                          = ( s @ A @ d ) ) ) ) )
                & ( ( ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                    & ! [I: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                          = ( s @ A @ d ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) ) ) )
          & ( ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                    & ! [I: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                          = ( s @ A @ d ) ) ) ) )
                & ( ( ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                    & ! [I: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                       => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) )
                          = ( s @ A @ d ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ A ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[730]) ).

thf(519,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ! [H: $i] :
                    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ H ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) @ ( s @ num @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ num @ H ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ num @ ( fun @ B @ bool ) ) ) @ D ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ B @ G ) ) ) )
            <=> ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) @ ( s @ num @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ B @ H ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ num @ ( fun @ B @ bool ) ) ) @ D ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_CART) ).

thf(6292,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ! [H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) @ ( s @ num @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ num @ H ) ) ) ) ) ) ) )
                & ( ! [H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) @ ( s @ num @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ num @ H ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ! [H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) @ ( s @ num @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ num @ H ) ) ) ) ) ) ) )
                & ( ! [H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) @ ( s @ num @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ num @ H ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ num @ ( fun @ B @ bool ) ) ) @ D ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ B @ G ) ) ) )
               => ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) @ ( s @ num @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ B @ H ) ) ) ) )
              & ( ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) @ ( s @ num @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ B @ H ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ num @ ( fun @ B @ bool ) ) ) @ D ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ num @ ( fun @ B @ bool ) ) ) @ D ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).

thf(1890,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( cart @ A @ B ) @ C )
        = ( s @ ( cart @ A @ B ) @ D ) )
    <=> ! [E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARTu_EQu_FULL) ).

thf(14681,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( cart @ A @ B ) @ C )
          = ( s @ ( cart @ A @ B ) @ D ) )
       => ! [E: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) )
      & ( ! [E: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) )
       => ( ( s @ ( cart @ A @ B ) @ C )
          = ( s @ ( cart @ A @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1890]) ).

thf(1602,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_BOUNDu_LTu_ALL) ).

thf(12881,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1602]) ).

thf(743,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        | ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_SUB) ).

thf(7625,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          | ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          | ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[743]) ).

thf(2740,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_WELLDEFu_LEMMA) ).

thf(19322,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2740]) ).

thf(1398,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) )
                   != ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asupport) ).

thf(11629,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) )
                     != ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) )
                     != ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) )
                     != ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) )
                     != ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1398]) ).

thf(2204,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_LSUB) ).

thf(16497,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2204]) ).

thf(2712,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_SYM) ).

thf(19190,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2712]) ).

thf(1070,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFACTu_LE) ).

thf(9391,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1070]) ).

thf(2889,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
     => ( ( s @ num @ A )
        = ( s @ num @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMPAIRu_INJu_LEMMA) ).

thf(20197,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
     => ( ( s @ num @ A )
        = ( s @ num @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2889]) ).

thf(1557,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_CIRCLE) ).

thf(12618,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1557]) ).

thf(731,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_PAIR) ).

thf(7545,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[731]) ).

thf(3021,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_INTERS) ).

thf(20876,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3021]) ).

thf(3178,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( p @ ( s @ bool @ A ) )
       => ~ ( p @ ( s @ bool @ B ) ) )
    <=> ( ( p @ ( s @ bool @ B ) )
       => ( p @ ( s @ bool @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONTRAPOSu_THM) ).

thf(21732,plain,
    ! [A: $i,B: $i] :
      ( ( ( ~ ( p @ ( s @ bool @ A ) )
         => ~ ( p @ ( s @ bool @ B ) ) )
       => ( ( p @ ( s @ bool @ B ) )
         => ( p @ ( s @ bool @ A ) ) ) )
      & ( ( ( p @ ( s @ bool @ B ) )
         => ( p @ ( s @ bool @ A ) ) )
       => ( ~ ( p @ ( s @ bool @ A ) )
         => ~ ( p @ ( s @ bool @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3178]) ).

thf(320,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ C )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_DIVu_LE) ).

thf(5113,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ C )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).

thf(481,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i,C: $i] :
        ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LINVu_LEMMA7) ).

thf(6096,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i,C: $i] :
        ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).

thf(293,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MUL) ).

thf(4979,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(2467,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
        <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ? [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ F ) ) ) ) )
        <=> ? [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_INu_GSPECu_conjunct0) ).

thf(17891,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
           => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) ) )
          & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ? [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ F ) ) ) ) )
           => ? [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) ) ) )
          & ( ? [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) ) )
           => ? [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2467]) ).

thf(3046,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_RADD) ).

thf(21066,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3046]) ).

thf(1597,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ B )
       != ( s @ ( cart @ real @ A ) @ C ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_POSu_LT) ).

thf(12849,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ B )
       != ( s @ ( cart @ real @ A ) @ C ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1597]) ).

thf(2266,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIFFu_EMPTY) ).

thf(16793,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2266]) ).

thf(797,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_RDISTRIB) ).

thf(7934,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[797]) ).

thf(1552,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_ADD) ).

thf(12603,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1552]) ).

thf(2158,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ B ) )
        & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_EXISTSu_ANDu_THM) ).

thf(16252,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ( p @ ( s @ bool @ B ) )
          & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ B ) )
          & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2158]) ).

thf(1453,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( list @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aAPPENDu_NIL) ).

thf(11948,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( list @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1453]) ).

thf(408,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
        <=> ? [G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
          = ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCROSS) ).

thf(5658,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
           => ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
          & ( ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
          = ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).

thf(781,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALLu_EL) ).

thf(7853,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[781]) ).

thf(842,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_MULu_LTu_LEMMA) ).

thf(8226,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[842]) ).

thf(1731,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_POS) ).

thf(13681,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1731]) ).

thf(3069,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
    <=> ( ( ( s @ A @ B )
          = ( s @ A @ C ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_INSERT) ).

thf(21187,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
       => ( ( ( s @ A @ B )
            = ( s @ A @ C ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ( ( ( s @ A @ B )
            = ( s @ A @ C ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3069]) ).

thf(812,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LBOUND) ).

thf(8031,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[812]) ).

thf(1142,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_ABSu_NUMSEG) ).

thf(9885,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1142]) ).

thf(1779,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_max) ).

thf(14100,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1779]) ).

thf(1406,axiom,
    ! [A: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
        = ( s @ int @ A ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_REFL) ).

thf(11702,plain,
    ! [A: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
          = ( s @ int @ A ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
          = ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1406]) ).

thf(1697,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ B ) @ num ) @ length ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) )
     => ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ B ) @ B ) ) @ el ) @ ( s @ num @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aELu_MAP) ).

thf(13515,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ B ) @ num ) @ length ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) )
     => ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ B ) @ B ) ) @ el ) @ ( s @ num @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1697]) ).

thf(86,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ARCHu_POW) ).

thf(3647,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(2508,axiom,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_ANTISYM) ).

thf(18108,plain,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2508]) ).

thf(3146,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ C ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',avec) ).

thf(21576,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ C ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3146]) ).

thf(771,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ H ) ) ) )
            <=> ? [I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ J ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ G ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ A @ I ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ H ) ) ) @ ( s @ B @ I ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) )
             => ! [F: $i] :
                  ( ! [G: $i,H: $i,I: $i] :
                      ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ ( fun @ B @ real ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) )
                      = ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) )
                 => ! [G: $i] :
                      ( ! [H: $i,I: $i,J: $i,K: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                        <=> ? [L: $i,M: $i] :
                              ( ( ( p @ ( s @ bool @ M ) )
                              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ J ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ K ) ) ) @ ( s @ bool @ M ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                     => ! [H: $i] :
                          ( ! [I: $i,J: $i,K: $i,L: $i] :
                              ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) ) @ H ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ K ) ) ) @ ( s @ A @ L ) ) )
                              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ A @ L ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ ( fun @ B @ real ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ K ) ) ) @ ( s @ A @ L ) ) ) ) ) )
                         => ! [I: $i,J: $i,K: $i,L: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) ) )
                             => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) ) @ H ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ J ) ) ) ) )
                                = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ J ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SUMu_RESTRICT) ).

thf(7767,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ H ) ) ) )
               => ? [I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ A @ I ) ) ) ) ) )
              & ( ? [I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ A @ I ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ H ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ H ) ) ) @ ( s @ B @ I ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) )
             => ! [F: $i] :
                  ( ! [G: $i,H: $i,I: $i] :
                      ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ ( fun @ B @ real ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) )
                      = ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) )
                 => ! [G: $i] :
                      ( ! [H: $i,I: $i,J: $i,K: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                           => ? [L: $i,M: $i] :
                                ( ( ( p @ ( s @ bool @ M ) )
                                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ J ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ J ) ) ) @ ( s @ B @ L ) ) ) ) )
                                 => ( p @ ( s @ bool @ M ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ K ) ) ) @ ( s @ bool @ M ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                          & ( ? [L: $i,M: $i] :
                                ( ( ( p @ ( s @ bool @ M ) )
                                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ J ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ J ) ) ) @ ( s @ B @ L ) ) ) ) )
                                 => ( p @ ( s @ bool @ M ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ K ) ) ) @ ( s @ bool @ M ) ) ) @ ( s @ B @ L ) ) ) ) )
                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) )
                     => ! [H: $i] :
                          ( ! [I: $i,J: $i,K: $i,L: $i] :
                              ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) ) @ H ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ K ) ) ) @ ( s @ A @ L ) ) )
                              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ A @ L ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ ( fun @ B @ real ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ K ) ) ) @ ( s @ A @ L ) ) ) ) ) )
                         => ! [I: $i,J: $i,K: $i,L: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) ) )
                             => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) ) @ H ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ J ) ) ) ) )
                                = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ B @ real ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ J ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[771]) ).

thf(148,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_LEu_SQUARE) ).

thf(3931,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(40,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) )
            & ! [E: $i,F: $i,G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ E ) ) ) @ ( s @ B @ G ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ F ) ) ) @ ( s @ B @ H ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                & ! [H: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_RELATED) ).

thf(3408,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) )
            & ! [E: $i,F: $i,G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ E ) ) ) @ ( s @ B @ G ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ F ) ) ) @ ( s @ B @ H ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                & ! [H: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(255,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_MUL) ).

thf(4680,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(871,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i,I: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ G ) ) ) @ ( s @ num @ H ) ) ) @ ( s @ num @ I ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ G ) ) ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ num @ H ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) ) ) @ E ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ F ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) ) @ ( s @ num @ I ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ C @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ F ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) ) @ ( s @ num @ I ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i,I: $i] :
                  ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( cart @ real @ B ) ) ) ) @ F ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ G ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) ) @ ( s @ num @ I ) ) )
                  = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) ) ) @ E ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ G ) ) ) @ ( s @ num @ I ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) ) ) ) )
             => ! [G: $i,H: $i] :
                  ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ G ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) )
                  = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( cart @ real @ B ) ) ) ) @ F ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ G ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amatrixu_mul) ).

thf(8403,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i,I: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ G ) ) ) @ ( s @ num @ H ) ) ) @ ( s @ num @ I ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ G ) ) ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ num @ H ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) ) ) @ E ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ F ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) ) @ ( s @ num @ I ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ C @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ F ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) ) @ ( s @ num @ I ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i,I: $i] :
                  ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( cart @ real @ B ) ) ) ) @ F ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ G ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) ) @ ( s @ num @ I ) ) )
                  = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ real ) ) ) ) @ E ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ G ) ) ) @ ( s @ num @ I ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) ) ) ) )
             => ! [G: $i,H: $i] :
                  ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ G ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) )
                  = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( fun @ num @ ( cart @ real @ B ) ) ) ) @ F ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ G ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ H ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[871]) ).

thf(354,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GEu_conjunct5) ).

thf(5330,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(1109,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) )
     => ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_STILLNZ) ).

thf(9641,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) )
     => ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1109]) ).

thf(2903,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ A @ B )
          = ( s @ A @ C ) )
        & ( ( s @ A @ C )
          = ( s @ A @ D ) ) )
     => ( ( s @ A @ B )
        = ( s @ A @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_TRANS) ).

thf(20244,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ A @ B )
          = ( s @ A @ C ) )
        & ( ( s @ A @ C )
          = ( s @ A @ D ) ) )
     => ( ( s @ A @ B )
        = ( s @ A @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2903]) ).

thf(121,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_SUB) ).

thf(3799,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(751,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        <=> ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_INTERS) ).

thf(7686,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) )
          & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[751]) ).

thf(170,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_CLAUSESu_conjunct4) ).

thf(4027,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(174,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ A )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_SUB) ).

thf(4063,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ A )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(2287,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_MUL) ).

thf(16899,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2287]) ).

thf(582,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_mul) ).

thf(6709,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).

thf(1587,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_LEu_1) ).

thf(12760,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1587]) ).

thf(1505,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ A ) )
    <=> ( ( s @ real @ B )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_ADDu_LCANCELu_0) ).

thf(12264,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ A ) )
       => ( ( s @ real @ B )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ real @ B )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1505]) ).

thf(1757,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ E ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ E ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_EQu_NUMSEG) ).

thf(13872,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ E ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ E ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1757]) ).

thf(878,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ D )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_LTu_ALL) ).

thf(8429,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ D )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).

thf(2672,axiom,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUPu_UNIQUE) ).

thf(18971,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
        = ( s @ real @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2672]) ).

thf(2027,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LTu_conjunct6) ).

thf(15559,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2027]) ).

thf(794,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
               => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_UNIONu_NONZERO) ).

thf(7923,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
               => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[794]) ).

thf(2526,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_EQu_RDOT) ).

thf(18204,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) )
       => ! [D: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2526]) ).

thf(397,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_CLAUSESu_conjunct1) ).

thf(5626,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).

thf(99,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ rows ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
          = ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arows) ).

thf(3699,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ rows ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
          = ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(901,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ B ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_ADDu_RCANCELu_0) ).

thf(8530,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ B ) )
       => ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[901]) ).

thf(772,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_CARDu_LT) ).

thf(7799,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[772]) ).

thf(1363,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
      <=> ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_CARDu_EQ) ).

thf(11356,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ C ) ) )
        & ( ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ C ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1363]) ).

thf(526,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_ADDu_LINV) ).

thf(6339,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).

thf(1411,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_BOUNDu_LTu_GEN) ).

thf(11734,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1411]) ).

thf(1619,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
        = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) )
    <=> ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_EQUALu_COLUMNS) ).

thf(12987,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
          = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) )
       => ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) )
      & ( ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
       => ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
          = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1619]) ).

thf(521,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ nil ) ) ) @ ( s @ C @ b0 ) ) )
      = ( s @ C @ b0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITLIST2u_conjunct0) ).

thf(6324,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ nil ) ) ) @ ( s @ C @ b0 ) ) )
      = ( s @ C @ b0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).

thf(2075,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
     => ? [C: $i] :
          ( ( s @ hreal @ B )
          = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_LEu_EXISTS) ).

thf(15739,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
     => ? [C: $i] :
          ( ( s @ hreal @ B )
          = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2075]) ).

thf(314,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_RINV) ).

thf(5053,plain,
    ! [A: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).

thf(334,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ num @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVISIONu_SIMPu_conjunct1) ).

thf(5221,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ num @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).

thf(1289,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amatrixu_add) ).

thf(10748,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1289]) ).

thf(905,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_EQu_0u_IMP) ).

thf(8545,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[905]) ).

thf(2376,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) )
     => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apairu_cases) ).

thf(17306,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) )
     => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2376]) ).

thf(1825,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
        & ! [F: $i,G: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ D ) @ ( s @ A @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ D ) @ ( s @ A @ G ) ) ) ) )
           => ( ( s @ A @ F )
              = ( s @ A @ G ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ o ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ A ) @ D ) ) ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_INJECTION) ).

thf(14327,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
        & ! [F: $i,G: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ D ) @ ( s @ A @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ D ) @ ( s @ A @ G ) ) ) ) )
           => ( ( s @ A @ F )
              = ( s @ A @ G ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ o ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ A ) @ D ) ) ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1825]) ).

thf(1522,axiom,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LEu_conjunct1) ).

thf(12404,plain,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1522]) ).

thf(2221,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GEu_conjunct1) ).

thf(16588,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2221]) ).

thf(2670,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ empty ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJOINTu_EMPTYu_REFL) ).

thf(18961,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2670]) ).

thf(1546,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_SUBu_LADD) ).

thf(12538,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1546]) ).

thf(2066,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_ADD) ).

thf(15714,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2066]) ).

thf(2831,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
    ! [D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ char @ A ) @ C ) @ ( s @ char @ ( i @ ( s @ ( fun @ bool @ char ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ char ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) ) ) @ ascii ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ bool @ K ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ bool @ A ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) ) ) ) ) ) @ B ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ bool @ K ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',acharu_RECURSION) ).

thf(19864,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
    ! [D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ char @ A ) @ C ) @ ( s @ char @ ( i @ ( s @ ( fun @ bool @ char ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ char ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) ) ) @ ascii ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ bool @ K ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ bool @ A ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ A ) ) ) ) ) ) ) ) @ B ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ bool @ K ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2831]) ).

thf(1668,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ C )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_MULTu_ADD) ).

thf(13347,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ C )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1668]) ).

thf(622,axiom,
    ! [A: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ hreal @ A ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_MULu_LZERO) ).

thf(6940,plain,
    ! [A: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ hreal @ A ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).

thf(1589,axiom,
    ! [A: $i] :
      ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_FORALLu_ABS) ).

thf(12792,plain,
    ! [A: $i] :
      ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
       => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) )
      & ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) )
       => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1589]) ).

thf(348,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ D ) ) )
     != ( s @ ( recspace @ A ) @ bottom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONSTRu_BOT) ).

thf(5270,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ D ) ) )
     != ( s @ ( recspace @ A ) @ bottom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(686,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_BASISu_1) ).

thf(7249,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[686]) ).

thf(1944,axiom,
    ( ( ( p @ ( s @ bool @ p1 ) )
      & ( p @ ( s @ bool @ p1 ) ) )
  <=> ( p @ ( s @ bool @ p1 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONJu_ACIu_conjunct3) ).

thf(15078,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1944]) ).

thf(525,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_POSu_LEu_NUMSEG) ).

thf(6335,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ D ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).

thf(2089,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LT2u_ODD) ).

thf(15847,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2089]) ).

thf(3001,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GTu_conjunct0) ).

thf(20787,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3001]) ).

thf(1319,axiom,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_DISTu_LEMMA) ).

thf(10986,plain,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1319]) ).

thf(1004,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( ( s @ num @ B )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_EQu_0) ).

thf(9073,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1004]) ).

thf(2002,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ univ ) @ ( s @ A @ B ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIV) ).

thf(15454,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ univ ) @ ( s @ A @ B ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2002]) ).

thf(3094,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_NEG) ).

thf(21325,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3094]) ).

thf(359,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_REALu_INTERVALu_conjunct5) ).

thf(5345,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).

thf(577,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ! [C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_0) ).

thf(6691,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ! [C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).

thf(2304,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
         != ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) )
     => ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSEGu_PROPERu_SUBSETu_FL) ).

thf(16968,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
         != ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) )
     => ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2304]) ).

thf(1298,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ) ) @ ( s @ real @ z ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ y ) ) ) @ ( s @ real @ z ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_ACIu_conjunct1) ).

thf(10791,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ) ) @ ( s @ real @ z ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ y ) ) ) @ ( s @ real @ z ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1298]) ).

thf(1441,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ bool ) @ num ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
            & ! [E: $i,F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
                  & ( ( s @ ( fun @ A @ bool ) @ E )
                   != ( s @ ( fun @ A @ bool ) @ F ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
               => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ G ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ bool ) @ num ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_UNIONSu_NONZERO) ).

thf(11884,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ bool ) @ num ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
            & ! [E: $i,F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
                  & ( ( s @ ( fun @ A @ bool ) @ E )
                   != ( s @ ( fun @ A @ bool ) @ F ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
               => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ G ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ bool ) @ num ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1441]) ).

thf(1215,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
            & ! [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                  & ( ( s @ B @ H )
                   != ( s @ B @ I ) )
                  & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ F ) @ ( s @ B @ H ) ) )
                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ F ) @ ( s @ B @ I ) ) ) ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ F ) @ ( s @ B @ H ) ) ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ C @ A ) @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ C @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_IMAGEu_NONZERO) ).

thf(10281,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
            & ! [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                  & ( ( s @ B @ H )
                   != ( s @ B @ I ) )
                  & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ F ) @ ( s @ B @ H ) ) )
                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ F ) @ ( s @ B @ I ) ) ) ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ F ) @ ( s @ B @ H ) ) ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ C @ A ) @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ C @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1215]) ).

thf(625,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_EQu_0) ).

thf(6955,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[625]) ).

thf(1033,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_SUPu_LE) ).

thf(9245,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1033]) ).

thf(3078,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_NEG) ).

thf(21234,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3078]) ).

thf(571,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
    <=> ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( s @ int @ B )
          = ( s @ int @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EQu_MULu_LCANCEL) ).

thf(6603,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
       => ( ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ int @ B )
            = ( s @ int @ C ) ) ) )
      & ( ( ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ int @ B )
            = ( s @ int @ C ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).

thf(1153,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_SUBu_RZERO) ).

thf(9945,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1153]) ).

thf(3135,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEUCLIDEANu_SPACEu_INFINITE) ).

thf(21533,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3135]) ).

thf(2983,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aGE) ).

thf(20678,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2983]) ).

thf(2228,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i] :
          ( ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ E ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ F ) @ ( s @ ( fun @ D @ E ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) )
          = ( s @ ( fun @ D @ E ) @ G ) )
     => ! [G: $i,H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ D @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ E ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ F ) @ ( s @ ( fun @ D @ E ) @ I ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_CONST) ).

thf(16629,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i] :
          ( ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ E ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ F ) @ ( s @ ( fun @ D @ E ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) )
          = ( s @ ( fun @ D @ E ) @ G ) )
     => ! [G: $i,H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ D @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ E ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) ) @ F ) @ ( s @ ( fun @ D @ E ) @ I ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2228]) ).

thf(541,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',adimindex) ).

thf(6459,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).

thf(2132,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_EQu_IMPu_LE) ).

thf(16108,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2132]) ).

thf(723,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_LZERO) ).

thf(7481,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[723]) ).

thf(2318,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_INu_CLAUSESu_conjunct1) ).

thf(17010,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          & ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          & ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) )
       => ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2318]) ).

thf(1260,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
     => ? [D: $i] :
        ! [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBOUNDSu_NOTZERO) ).

thf(10566,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
     => ? [D: $i] :
        ! [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1260]) ).

thf(270,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ s0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERSu_1) ).

thf(4789,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ s0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(2928,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_TRANS) ).

thf(20387,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2928]) ).

thf(1323,axiom,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y1 ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y2 ) ) ) ) )
   => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y2 ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y2 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRATu_LEMMA4) ).

thf(11007,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y1 ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y2 ) ) ) ) )
   => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y2 ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y2 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1323]) ).

thf(247,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
          = ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZE) ).

thf(4636,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
            = ( s @ num @ C ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
            = ( s @ num @ C ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(818,axiom,
    ! [A: $i] :
      ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ? [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EXISTSu_POS) ).

thf(8063,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
       => ? [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ B ) ) ) ) ) )
      & ( ? [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ B ) ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[818]) ).

thf(1271,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_LMULu_EQ) ).

thf(10644,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1271]) ).

thf(1941,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_LADDu_IMP) ).

thf(15065,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1941]) ).

thf(778,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) )
     => ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCHOOSEu_SUBSETu_STRONG) ).

thf(7838,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) )
     => ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[778]) ).

thf(1746,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_3) ).

thf(13825,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1746]) ).

thf(810,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i,G: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( ( s @ A @ F )
               != ( s @ A @ G ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ C ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) ) @ o ) @ ( s @ ( fun @ B @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_IMAGEu_NONZERO) ).

thf(8014,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [F: $i,G: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( ( s @ A @ F )
               != ( s @ A @ G ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ C ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) ) @ o ) @ ( s @ ( fun @ B @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[810]) ).

thf(741,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) ) @ F ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ H ) ) ) )
        <=> ! [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ E ) @ ( fun @ ( cart @ real @ E ) @ bool ) ) @ geq ) @ ( s @ ( cart @ real @ E ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ H ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ E ) @ ( i @ ( s @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ f ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( cart @ real @ D ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ D ) ) @ g ) @ ( s @ B @ J ) ) ) ) ) ) ) ) )
     => ! [G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) )
         => ( ( s @ ( cart @ real @ E ) @ ( i @ ( s @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ f ) ) ) ) ) @ ( s @ ( cart @ real @ D ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ D ) ) @ ( cart @ real @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ D ) ) @ ( cart @ real @ D ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ D ) ) @ g ) ) ) ) )
            = ( s @ ( cart @ real @ E ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ ( cart @ real @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ ( cart @ real @ E ) ) ) @ vsum ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) ) @ F ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_VSUM) ).

thf(7610,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) ) @ F ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ H ) ) ) )
           => ! [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ E ) @ ( fun @ ( cart @ real @ E ) @ bool ) ) @ geq ) @ ( s @ ( cart @ real @ E ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ H ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ E ) @ ( i @ ( s @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ f ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( cart @ real @ D ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ D ) ) @ g ) @ ( s @ B @ J ) ) ) ) ) ) ) ) )
          & ( ! [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ E ) @ ( fun @ ( cart @ real @ E ) @ bool ) ) @ geq ) @ ( s @ ( cart @ real @ E ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ H ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ E ) @ ( i @ ( s @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ f ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( cart @ real @ D ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ D ) ) @ g ) @ ( s @ B @ J ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) ) @ F ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ H ) ) ) ) ) )
     => ! [G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) )
         => ( ( s @ ( cart @ real @ E ) @ ( i @ ( s @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ f ) ) ) ) ) @ ( s @ ( cart @ real @ D ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ D ) ) @ ( cart @ real @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ D ) ) @ ( cart @ real @ D ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ D ) ) @ g ) ) ) ) )
            = ( s @ ( cart @ real @ E ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ ( cart @ real @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ ( cart @ real @ E ) ) ) @ vsum ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ E ) ) @ bool ) ) @ F ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ D ) @ ( cart @ real @ E ) ) ) @ G ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[741]) ).

thf(3087,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_ADD2) ).

thf(21275,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3087]) ).

thf(2494,axiom,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LETu_ANTISYM) ).

thf(18017,plain,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2494]) ).

thf(1101,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ x ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_conjunct1) ).

thf(9576,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ x ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ x ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1101]) ).

thf(2959,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_ge) ).

thf(20553,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2959]) ).

thf(393,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_DIFF) ).

thf(5583,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).

thf(157,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_DIVu_POW2) ).

thf(3963,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(2938,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_LE) ).

thf(20448,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2938]) ).

thf(2562,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_SUBu_LADD) ).

thf(18404,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2562]) ).

thf(1090,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ n3 @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ n3 @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_3) ).

thf(9525,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ n3 @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ n3 @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1090]) ).

thf(3062,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
        <=> ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ C ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ) ) )
          = ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIREDu_ETAu_THMu_conjunct0) ).

thf(21150,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
           => ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) ) )
          & ( ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) ) ) )
     => ! [E: $i] :
          ( ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ C ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ) ) )
          = ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3062]) ).

thf(752,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_RESTRICT) ).

thf(7698,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[752]) ).

thf(106,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_BOUND) ).

thf(3746,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(2963,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ inja ) @ ( s @ A @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) )
    <=> ( ( s @ A @ D )
        = ( s @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJA) ).

thf(20570,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ inja ) @ ( s @ A @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) )
       => ( ( s @ A @ D )
          = ( s @ A @ B ) ) )
      & ( ( ( s @ A @ D )
          = ( s @ A @ B ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ inja ) @ ( s @ A @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2963]) ).

thf(531,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_ACIu_conjunct1) ).

thf(6360,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).

thf(1449,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ int @ B )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_RNEGu_UNIQ) ).

thf(11929,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ int @ B )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      & ( ( ( s @ int @ B )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1449]) ).

thf(1371,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) )
      & ? [D: $i,E: $i] :
          ( ( s @ int @ C )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ D ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_GCDu_EXISTSu_POS) ).

thf(11400,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) )
      & ? [D: $i,E: $i] :
          ( ( s @ int @ C )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ D ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1371]) ).

thf(2745,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] : ( p @ ( s @ bool @ B ) )
    <=> ( p @ ( s @ bool @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_SIMP) ).

thf(19341,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2745]) ).

thf(87,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
         => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ p0 ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
    <=> ? [C: $i] :
        ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ p0 ) @ ( s @ num @ D ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLAMBDAu_SKOLEM) ).

thf(3650,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ p0 ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
       => ? [C: $i] :
          ! [D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ p0 ) @ ( s @ num @ D ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) )
      & ( ? [C: $i] :
          ! [D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ p0 ) @ ( s @ num @ D ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) )
       => ! [C: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ p0 ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(1631,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( ? [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) )
                    = ( s @ A @ F ) ) )
              & ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) )
                      = ( s @ A @ F ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) )
                 => ( ( s @ B @ G )
                    = ( s @ B @ H ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ E ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_IMAGEu_INJu_EQ) ).

thf(13074,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( ? [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) )
                    = ( s @ A @ F ) ) )
              & ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) )
                      = ( s @ A @ F ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) )
                 => ( ( s @ B @ G )
                    = ( s @ B @ H ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ E ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1631]) ).

thf(3110,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_ADDu_WELLDEFR) ).

thf(21421,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3110]) ).

thf(920,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',adist) ).

thf(8662,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[920]) ).

thf(2071,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aAPPENDu_ASSOC) ).

thf(15729,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2071]) ).

thf(1815,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) @ ( s @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_SING) ).

thf(14276,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) @ ( s @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1815]) ).

thf(2126,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_LNEG) ).

thf(16055,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2126]) ).

thf(47,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_POW2) ).

thf(3440,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(1292,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_UNIONSu_conjunct1) ).

thf(10760,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1292]) ).

thf(1979,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ i1 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_I) ).

thf(15281,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ i1 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1979]) ).

thf(828,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECu_COMPONENT) ).

thf(8138,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[828]) ).

thf(371,axiom,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ u_0 ) ) )
    = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EVENu_conjunct1) ).

thf(5475,plain,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ u_0 ) ) )
    = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).

thf(452,axiom,
    ! [A: $i] :
      ( ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_CASES) ).

thf(5951,plain,
    ! [A: $i] :
      ( ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).

thf(1618,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ! [G: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_LISTu_GEN) ).

thf(12974,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
                & ( ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
                & ( ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1618]) ).

thf(1781,axiom,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_REFL) ).

thf(14110,plain,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1781]) ).

thf(2607,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_NEGu_NEG) ).

thf(18672,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2607]) ).

thf(1093,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        <=> ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ x ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_inv) ).

thf(9530,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
           => ? [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) )
          & ( ? [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ x ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1093]) ).

thf(197,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ B ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ B @ D ) )
     => ! [D: $i,E: $i] :
        ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
          <=> ( ( s @ ( fun @ A @ bool ) @ D )
              = ( s @ ( fun @ A @ bool ) @ empty ) ) )
          & ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ B ) ) @ C ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
            = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_CONST) ).

thf(4291,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ B ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ B @ D ) )
     => ! [D: $i,E: $i] :
        ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
           => ( ( s @ ( fun @ A @ bool ) @ D )
              = ( s @ ( fun @ A @ bool ) @ empty ) ) )
          & ( ( ( s @ ( fun @ A @ bool ) @ D )
              = ( s @ ( fun @ A @ bool ) @ empty ) )
           => ( p @ ( s @ bool @ F ) ) )
          & ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ B ) ) @ C ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
            = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(2790,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ E ) ) ) @ ( s @ ( list @ A ) @ F ) ) ) ) ) @ ( s @ B @ D ) ) )
      = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( list @ A ) @ F ) ) ) @ ( s @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITLISTu_APPEND) ).

thf(19620,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ E ) ) ) @ ( s @ ( list @ A ) @ F ) ) ) ) ) @ ( s @ B @ D ) ) )
      = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( list @ A ) @ ( fun @ B @ B ) ) ) @ itlist ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( list @ A ) @ F ) ) ) @ ( s @ B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2790]) ).

thf(2884,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ A )
        = ( s @ int @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EQu_IMPu_LE) ).

thf(20175,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ A )
        = ( s @ int @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2884]) ).

thf(834,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_SING) ).

thf(8176,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[834]) ).

thf(1875,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_WELLDEF) ).

thf(14623,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1875]) ).

thf(154,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ n10 @ bool ) @ A ) @ ( s @ n10 @ B ) ) )
          = ( s @ bool @ t ) )
     => ( ( s @ n10 @ one )
        = ( s @ n10 @ ( i @ ( s @ ( fun @ ( fun @ n10 @ bool ) @ n10 ) @ h_ ) @ ( s @ ( fun @ n10 @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aoneu_DEF) ).

thf(3954,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ n10 @ bool ) @ A ) @ ( s @ n10 @ B ) ) )
          = ( s @ bool @ t ) )
     => ( ( s @ n10 @ one )
        = ( s @ n10 @ ( i @ ( s @ ( fun @ ( fun @ n10 @ bool ) @ n10 ) @ h_ ) @ ( s @ ( fun @ n10 @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(1763,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ARCHu_POWu_INV) ).

thf(13928,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1763]) ).

thf(2382,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ B @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ B @ F ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
            <=> ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ E ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ A @ H ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i] :
                ? [G: $i] :
                  ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) )
                  = ( s @ B @ F ) )
            <=> ! [F: $i] :
                  ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ) )
                  = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJECTIVEu_IMAGEu_THM) ).

thf(17328,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ B @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ B @ F ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ B @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
               => ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ E ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ A @ H ) ) ) ) )
              & ( ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ E ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ A @ H ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ( ! [F: $i] :
                  ? [G: $i] :
                    ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ F ) )
               => ! [F: $i] :
                    ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ) )
                    = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) )
              & ( ! [F: $i] :
                    ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ) )
                    = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
               => ! [F: $i] :
                  ? [G: $i] :
                    ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2382]) ).

thf(790,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMu_LISTu_OFu_SET) ).

thf(7898,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[790]) ).

thf(153,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_LMUL) ).

thf(3952,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(3037,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( s @ A @ C )
          = ( s @ A @ B ) )
      & ! [C: $i,D: $i] :
          ( ( ( ( s @ A @ C )
              = ( s @ A @ B ) )
            & ( ( s @ A @ D )
              = ( s @ A @ B ) ) )
         => ( ( s @ A @ C )
            = ( s @ A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_UNIQUEu_REFL) ).

thf(20979,plain,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( s @ A @ C )
          = ( s @ A @ B ) )
      & ! [C: $i,D: $i] :
          ( ( ( ( s @ A @ C )
              = ( s @ A @ B ) )
            & ( ( s @ A @ D )
              = ( s @ A @ B ) ) )
         => ( ( s @ A @ C )
            = ( s @ A @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3037]) ).

thf(1794,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_INVu_WELLDEF) ).

thf(14142,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1794]) ).

thf(199,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_0) ).

thf(4301,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(2083,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ A ) @ ( s @ ( fun @ num @ bool ) @ B ) ) ) @ ( s @ num @ C ) ) ) )
        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ C ) ) ) )
            & ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) )
               => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ D ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ minimal ) @ ( s @ ( fun @ num @ bool ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ h_ ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ A ) @ ( s @ ( fun @ num @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aminimal) ).

thf(15798,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ A ) @ ( s @ ( fun @ num @ bool ) @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ C ) ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) )
                 => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ D ) ) ) ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ C ) ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) )
                 => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ D ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ A ) @ ( s @ ( fun @ num @ bool ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ minimal ) @ ( s @ ( fun @ num @ bool ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ h_ ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ A ) @ ( s @ ( fun @ num @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2083]) ).

thf(2275,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EQu_conjunct0) ).

thf(16847,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2275]) ).

thf(1315,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_TRIVu_NUMSEG) ).

thf(10946,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1315]) ).

thf(2022,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_INSERT) ).

thf(15545,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2022]) ).

thf(2759,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_ADDu_RCANCEL) ).

thf(19425,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
       => ( ( s @ real @ A )
          = ( s @ real @ B ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ B ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2759]) ).

thf(628,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_0) ).

thf(6967,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[628]) ).

thf(426,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ A ) @ fst ) @ ( s @ ( prod @ A @ B ) @ C ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ B ) @ snd ) @ ( s @ ( prod @ A @ B ) @ C ) ) ) ) )
      = ( s @ ( prod @ A @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIR) ).

thf(5818,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ A ) @ fst ) @ ( s @ ( prod @ A @ B ) @ C ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ B ) @ snd ) @ ( s @ ( prod @ A @ B ) @ C ) ) ) ) )
      = ( s @ ( prod @ A @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).

thf(2094,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_NEG) ).

thf(15882,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2094]) ).

thf(2604,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SGNu_INV) ).

thf(18653,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2604]) ).

thf(1150,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ m ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_ACu_conjunct0) ).

thf(9910,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1150]) ).

thf(908,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LT) ).

thf(8555,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[908]) ).

thf(1,conjecture,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ matrix ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MUL) ).

thf(2,negated_conjecture,
    ~ ! [A: $i,B: $i,C: $i] :
        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
       => ! [D: $i] :
            ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ matrix ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(3179,plain,
    ~ ! [A: $i,B: $i,C: $i] :
        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
       => ! [D: $i] :
            ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ matrix ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(1617,axiom,
    ! [A: $i] :
      ( ( ( s @ int @ A )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_NZ) ).

thf(12967,plain,
    ! [A: $i] :
      ( ( ( ( s @ int @ A )
         != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) )
       => ( ( s @ int @ A )
         != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1617]) ).

thf(1868,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_CMULu_ASSOC) ).

thf(14566,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1868]) ).

thf(1659,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_LT) ).

thf(13291,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1659]) ).

thf(1906,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSsrnato_lequ_addr) ).

thf(14809,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1906]) ).

thf(2843,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ columns ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_COLUMNS) ).

thf(19921,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ columns ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2843]) ).

thf(757,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_MULu_LE) ).

thf(7717,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[757]) ).

thf(405,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_DIFF) ).

thf(5651,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).

thf(1846,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ A @ E )
           != ( s @ A @ F ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [E: $i,F: $i,G: $i,H: $i,I: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ B @ I ) ) ) @ ( s @ num @ F ) ) ) ) )
         => ( ( ( s @ B @ H )
              = ( s @ B @ I ) )
            & ( ( s @ num @ E )
              = ( s @ num @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINRECu_UNIQUEu_LEMMA) ).

thf(14473,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ A @ E )
           != ( s @ A @ F ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [E: $i,F: $i,G: $i,H: $i,I: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ B @ I ) ) ) @ ( s @ num @ F ) ) ) ) )
         => ( ( ( s @ B @ H )
              = ( s @ B @ I ) )
            & ( ( s @ num @ E )
              = ( s @ num @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1846]) ).

thf(10,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) ) @ ( s @ C @ b0 ) ) )
      = ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ B ) @ B ) @ hd ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ tl ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) @ ( s @ C @ b0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITLIST2u_DEFu_conjunct1) ).

thf(3210,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) ) @ ( s @ C @ b0 ) ) )
      = ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ B ) @ B ) @ hd ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ tl ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) @ ( s @ C @ b0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(2314,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_RNEG) ).

thf(17001,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2314]) ).

thf(1956,axiom,
    ! [A: $i] :
      ( ( ( ( p @ ( s @ bool @ t ) )
          & ( p @ ( s @ bool @ A ) ) )
      <=> ( p @ ( s @ bool @ A ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ t ) ) )
      <=> ( p @ ( s @ bool @ A ) ) )
      & ( ( ( p @ ( s @ bool @ f0 ) )
          & ( p @ ( s @ bool @ A ) ) )
      <=> ( p @ ( s @ bool @ f0 ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ f0 ) ) )
      <=> ( p @ ( s @ bool @ f0 ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ A ) ) )
      <=> ( p @ ( s @ bool @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aANDu_CLAUSES) ).

thf(15141,plain,
    ! [A: $i] :
      ( ( ( ( p @ ( s @ bool @ t ) )
          & ( p @ ( s @ bool @ A ) ) )
       => ( p @ ( s @ bool @ A ) ) )
      & ( ( p @ ( s @ bool @ A ) )
       => ( ( p @ ( s @ bool @ t ) )
          & ( p @ ( s @ bool @ A ) ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ t ) ) )
       => ( p @ ( s @ bool @ A ) ) )
      & ( ( p @ ( s @ bool @ A ) )
       => ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ t ) ) ) )
      & ( ( ( p @ ( s @ bool @ f0 ) )
          & ( p @ ( s @ bool @ A ) ) )
       => ( p @ ( s @ bool @ f0 ) ) )
      & ( ( p @ ( s @ bool @ f0 ) )
       => ( ( p @ ( s @ bool @ f0 ) )
          & ( p @ ( s @ bool @ A ) ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ f0 ) ) )
       => ( p @ ( s @ bool @ f0 ) ) )
      & ( ( p @ ( s @ bool @ f0 ) )
       => ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ f0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1956]) ).

thf(780,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
      <=> ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_CLAUSESu_NUMSEGu_conjunct0) ).

thf(7846,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
       => ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ C ) ) )
      & ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[780]) ).

thf(805,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ARCHu_ZERO) ).

thf(7981,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[805]) ).

thf(184,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ ( cart @ real @ n3 ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ n3 ) @ x ) ) ) @ ( s @ ( cart @ real @ n3 ) @ y ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_3) ).

thf(4167,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ ( cart @ real @ n3 ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ n3 ) @ x ) ) ) @ ( s @ ( cart @ real @ n3 ) @ y ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n3 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n3 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(1253,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
    <=> ? [C: $i] :
        ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anaddu_eq) ).

thf(10528,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
       => ? [C: $i] :
          ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) )
      & ( ? [C: $i] :
          ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1253]) ).

thf(1801,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ y ) ) ) @ ( s @ real @ x ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_ACIu_conjunct0) ).

thf(14212,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ y ) ) ) @ ( s @ real @ x ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1801]) ).

thf(532,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LEu_conjunct7) ).

thf(6363,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).

thf(2148,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_SUBSET) ).

thf(16201,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2148]) ).

thf(1640,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_MONOu_LTu_EQ) ).

thf(13190,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1640]) ).

thf(2818,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ A ) )
       => ( p @ ( s @ bool @ B ) ) )
    <=> ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ B ) ) )
      <=> ( p @ ( s @ bool @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMPu_DEF) ).

thf(19758,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ B ) ) )
       => ( ( ( ( p @ ( s @ bool @ A ) )
              & ( p @ ( s @ bool @ B ) ) )
           => ( p @ ( s @ bool @ A ) ) )
          & ( ( p @ ( s @ bool @ A ) )
           => ( ( p @ ( s @ bool @ A ) )
              & ( p @ ( s @ bool @ B ) ) ) ) ) )
      & ( ( ( ( ( p @ ( s @ bool @ A ) )
              & ( p @ ( s @ bool @ B ) ) )
           => ( p @ ( s @ bool @ A ) ) )
          & ( ( p @ ( s @ bool @ A ) )
           => ( ( p @ ( s @ bool @ A ) )
              & ( p @ ( s @ bool @ B ) ) ) ) )
       => ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2818]) ).

thf(2291,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ C ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) ) ) @ ( s @ num @ E ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i,G: $i,H: $i] :
                  ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) ) @ D ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ G ) ) ) @ ( s @ num @ H ) ) )
                  = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ G ) @ ( s @ num @ H ) ) ) ) ) )
             => ! [E: $i,F: $i,G: $i,H: $i,I: $i] :
                  ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) ) @ D ) @ ( s @ num @ G ) ) ) @ ( s @ num @ H ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ I ) ) ) ) )
                  = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ H ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) ) @ D ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ I ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_SWAPu_NUMSEG) ).

thf(16920,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ C ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) ) ) @ ( s @ num @ E ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i,G: $i,H: $i] :
                  ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) ) @ D ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ G ) ) ) @ ( s @ num @ H ) ) )
                  = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ G ) @ ( s @ num @ H ) ) ) ) ) )
             => ! [E: $i,F: $i,G: $i,H: $i,I: $i] :
                  ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) ) @ D ) @ ( s @ num @ G ) ) ) @ ( s @ num @ H ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ I ) ) ) ) )
                  = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ H ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) ) @ D ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ I ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2291]) ).

thf(1221,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_LID) ).

thf(10342,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1221]) ).

thf(809,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_MULT) ).

thf(8008,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[809]) ).

thf(1077,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ B @ E )
           != ( s @ B @ F ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ A @ G ) ) ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ G ) ) ) ) ) ) )
     => ? [E: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) )
            = ( s @ A @ D ) )
          & ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
             => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ delete ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINRECu_FUN) ).

thf(9419,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ B @ E )
           != ( s @ B @ F ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ A @ G ) ) ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ G ) ) ) ) ) ) )
     => ? [E: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) )
            = ( s @ A @ D ) )
          & ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
             => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ delete ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1077]) ).

thf(2302,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MINu_LE) ).

thf(16958,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2302]) ).

thf(2750,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        | ( p @ ( s @ bool @ C ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_ORu_FORALLu_THM) ).

thf(19373,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ( p @ ( s @ bool @ C ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            | ( p @ ( s @ bool @ C ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            | ( p @ ( s @ bool @ C ) ) )
       => ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ( p @ ( s @ bool @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2750]) ).

thf(2217,axiom,
    ? [A: $i] : ( p @ ( s @ bool @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_ONEu_REP) ).

thf(16568,plain,
    ? [A: $i] : ( p @ ( s @ bool @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2217]) ).

thf(850,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LEu_conjunct3) ).

thf(8258,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[850]) ).

thf(221,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
        = ( s @ int @ B ) )
    <=> ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EQu_ADDu_RCANCELu_0) ).

thf(4421,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ B ) )
       => ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(2327,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ g_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',agtu_c) ).

thf(17074,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ g_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2327]) ).

thf(2040,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ! [E: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
                 => ( ( s @ ( fun @ A @ bool ) @ E )
                    = ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aKL) ).

thf(15619,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ! [E: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
                 => ( ( s @ ( fun @ A @ bool ) @ E )
                    = ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2040]) ).

thf(1928,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMERAL) ).

thf(15022,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1928]) ).

thf(342,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_ADDu_ABSORB) ).

thf(5253,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).

thf(739,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_CLAUSESu_conjunct0) ).

thf(7604,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[739]) ).

thf(2358,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) )
    <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_lt) ).

thf(17218,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2358]) ).

thf(1628,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ A ) @ B ) ) )
      = ( s @ ( list @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aAPPENDu_conjunct0) ).

thf(13066,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ A ) @ B ) ) )
      = ( s @ ( list @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1628]) ).

thf(234,axiom,
    ! [A: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ hreal @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_ADDu_RID) ).

thf(4513,plain,
    ! [A: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ hreal @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(2237,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_INV2) ).

thf(16656,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) )
       => ( ( s @ real @ A )
          = ( s @ real @ B ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ B ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2237]) ).

thf(2503,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_LEu_ADD2) ).

thf(18055,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2503]) ).

thf(816,axiom,
    ( ( ( ( p @ ( s @ bool @ a ) )
       => ( p @ ( s @ bool @ b ) ) )
      & ( ( p @ ( s @ bool @ c0 ) )
       => ( p @ ( s @ bool @ d0 ) ) ) )
   => ( ( ( p @ ( s @ bool @ a ) )
        & ( p @ ( s @ bool @ c0 ) ) )
     => ( ( p @ ( s @ bool @ b ) )
        & ( p @ ( s @ bool @ d0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOu_AND) ).

thf(8053,plain,
    ( ( ( ( p @ ( s @ bool @ a ) )
       => ( p @ ( s @ bool @ b ) ) )
      & ( ( p @ ( s @ bool @ c0 ) )
       => ( p @ ( s @ bool @ d0 ) ) ) )
   => ( ( ( p @ ( s @ bool @ a ) )
        & ( p @ ( s @ bool @ c0 ) ) )
     => ( ( p @ ( s @ bool @ b ) )
        & ( p @ ( s @ bool @ d0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[816]) ).

thf(1592,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
        <=> ? [G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ! [I: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
          = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERSu_IMAGE) ).

thf(12803,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
           => ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ! [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ! [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
          & ( ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ! [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ! [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
          = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1592]) ).

thf(331,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_DIFF) ).

thf(5213,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).

thf(1202,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        <=> ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_POWERSET) ).

thf(10195,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
          & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1202]) ).

thf(43,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_INV) ).

thf(3428,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(977,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ A ) ) ) @ ( s @ ( fun @ num @ bool ) @ empty ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_SING) ).

thf(8968,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ A ) ) ) @ ( s @ ( fun @ num @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[977]) ).

thf(1974,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_RNEG) ).

thf(15270,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1974]) ).

thf(604,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_ADDR) ).

thf(6837,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).

thf(2346,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( list @ char ) @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ ( list @ char ) @ bool ) @ univ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',astringu_INFINITE) ).

thf(17165,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( list @ char ) @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ ( list @ char ) @ bool ) @ univ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2346]) ).

thf(1327,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_PREu_conjunct1) ).

thf(11026,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1327]) ).

thf(2446,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAXu_LT) ).

thf(17780,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2446]) ).

thf(3091,axiom,
    ( ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ A ) ) ) )
  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ t ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ f0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_BOOLu_THM) ).

thf(21306,plain,
    ( ( ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ A ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ t ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ f0 ) ) ) ) ) )
    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ t ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ f0 ) ) ) ) )
     => ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3091]) ).

thf(2245,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        | ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_INu_INSERT) ).

thf(16683,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          | ? [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          | ? [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) )
       => ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2245]) ).

thf(711,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ADDu_conjunct9) ).

thf(7404,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[711]) ).

thf(2749,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ g ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) ) ) )
            = ( s @ B @ C ) ) )
    <=> ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
            & ( ( s @ A @ C )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ g ) @ ( s @ A @ C ) ) )
            = ( s @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFLATTENu_LEMMA) ).

thf(19358,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ g ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) ) ) )
              = ( s @ B @ C ) ) )
       => ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
              & ( ( s @ A @ C )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ g ) @ ( s @ A @ C ) ) )
              = ( s @ B @ D ) ) ) )
      & ( ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
              & ( ( s @ A @ C )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ g ) @ ( s @ A @ C ) ) )
              = ( s @ B @ D ) ) )
       => ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ g ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) ) ) )
              = ( s @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2749]) ).

thf(2191,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_BETWEEN1) ).

thf(16451,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2191]) ).

thf(696,axiom,
    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) ) ) )
  <=> ? [A: $i] :
        ( ( ( ( s @ num @ a0 )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ A ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) )
            & ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_ELIMu_THMi_) ).

thf(7313,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) ) ) )
     => ? [A: $i] :
          ( ( ( ( s @ num @ a0 )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ A ) ) ) )
            | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) )
              & ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) )
    & ( ? [A: $i] :
          ( ( ( ( s @ num @ a0 )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ A ) ) ) )
            | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) )
              & ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[696]) ).

thf(2603,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_NORMu_TRIANGLE) ).

thf(18650,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ real ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2603]) ).

thf(1789,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ A ) )
        & ( p @ ( s @ bool @ B ) )
        & ( p @ ( s @ bool @ C ) ) )
    <=> ( ( p @ ( s @ bool @ A ) )
        & ( p @ ( s @ bool @ B ) )
        & ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONJu_ASSOC) ).

thf(14131,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1789]) ).

thf(2361,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ union ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_UNION) ).

thf(17239,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ union ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2361]) ).

thf(2832,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ F ) ) ) )
                 => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ A @ F ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) )
         => ! [D: $i,E: $i] :
              ( ( ! [F: $i] :
                    ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ A @ F ) ) ) )
                & ! [F: $i] :
                    ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) )
             => ( ( s @ ( fun @ A @ B ) @ D )
                = ( s @ ( fun @ A @ B ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_UREC) ).

thf(19867,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ F ) ) ) )
                 => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ A @ F ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) )
         => ! [D: $i,E: $i] :
              ( ( ! [F: $i] :
                    ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ A @ F ) ) ) )
                & ! [F: $i] :
                    ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) )
             => ( ( s @ ( fun @ A @ B ) @ D )
                = ( s @ ( fun @ A @ B ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2832]) ).

thf(1518,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
    <=> ? [B: $i] :
          ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aODDu_EXISTS) ).

thf(12357,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
       => ? [B: $i] :
            ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      & ( ? [B: $i] :
            ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1518]) ).

thf(1078,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTER) ).

thf(9429,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1078]) ).

thf(1188,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
    <=> ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( s @ real @ B )
          = ( s @ real @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_MULu_LCANCEL) ).

thf(10094,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
       => ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ real @ B )
            = ( s @ real @ C ) ) ) )
      & ( ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ real @ B )
            = ( s @ real @ C ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1188]) ).

thf(971,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_LDISTRIB) ).

thf(8952,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[971]) ).

thf(268,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
            <=> ? [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ H ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i,G: $i,H: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ F ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [E: $i,F: $i,G: $i,H: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ G ) ) ) @ ( s @ ( fun @ A @ real ) @ H ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ G ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ H ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_CASES) ).

thf(4736,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
          & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
               => ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
              & ( ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i,G: $i,H: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ F ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [E: $i,F: $i,G: $i,H: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ G ) ) ) @ ( s @ ( fun @ A @ real ) @ H ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ G ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ H ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(1110,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) ) )
      <=> ( ( s @ num @ B )
         != ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_BASISu_BASIS) ).

thf(9645,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) ) )
         => ( ( s @ num @ B )
           != ( s @ num @ C ) ) )
        & ( ( ( s @ num @ B )
           != ( s @ num @ C ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1110]) ).

thf(1551,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_ONE) ).

thf(12600,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1551]) ).

thf(1516,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_RINV) ).

thf(12314,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1516]) ).

thf(2514,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ x ) ) )
    = ( s @ real @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_ACIu_conjunct3) ).

thf(18137,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ x ) ) )
    = ( s @ real @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2514]) ).

thf(2091,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ inj ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        & ! [F: $i,G: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
           => ( ( s @ B @ F )
              = ( s @ B @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJ) ).

thf(15852,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ inj ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
          & ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ B @ F )
                = ( s @ B @ G ) ) ) ) )
      & ( ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
          & ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ B @ F )
                = ( s @ B @ G ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ inj ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2091]) ).

thf(238,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_RZERO) ).

thf(4530,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(585,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SUPERSET) ).

thf(6724,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).

thf(2841,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
        <=> ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) )
         => ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ F ) ) )
            = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_EQ) ).

thf(19906,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) )
          & ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) )
         => ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ F ) ) )
            = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2841]) ).

thf(1866,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LTu_TRANS) ).

thf(14561,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1866]) ).

thf(3042,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SGNu_DIV) ).

thf(21006,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3042]) ).

thf(1258,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_SUMu_NUMSEG) ).

thf(10560,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1258]) ).

thf(1832,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) )
    <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_conjunct1) ).

thf(14369,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ) )
      & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1832]) ).

thf(3138,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
        ? [G: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ G ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) )
    <=> ? [F: $i] :
          ( ( s @ ( fun @ B @ A ) @ D )
          = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ C @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFUNCTIONu_FACTORSu_RIGHT) ).

thf(21543,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] :
          ? [G: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ G ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) )
       => ? [F: $i] :
            ( ( s @ ( fun @ B @ A ) @ D )
            = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ C @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) ) )
      & ( ? [F: $i] :
            ( ( s @ ( fun @ B @ A ) @ D )
            = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ C @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) )
       => ! [F: $i] :
          ? [G: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ G ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3138]) ).

thf(1195,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                    = ( s @ A @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ real ) @ G ) ) ) @ ( s @ A @ H ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) )
             => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_GROUP) ).

thf(10158,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                      = ( s @ A @ F ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ real ) @ G ) ) ) @ ( s @ A @ H ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) )
             => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ real ) @ ( fun @ A @ real ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1195]) ).

thf(1080,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ? [C: $i,D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ F ) ) ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ F ) ) ) ) )
             => ( ( s @ num @ E )
                = ( s @ num @ F ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) )
          & ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ D ) ) ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_INDEXu_NUMBERS) ).

thf(9461,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ? [C: $i,D: $i] :
            ( ! [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ F ) ) ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ F ) ) ) ) )
               => ( ( s @ num @ E )
                  = ( s @ num @ F ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) )
            & ( ( s @ ( fun @ A @ bool ) @ B )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ D ) ) ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) ) ) )
      & ( ? [C: $i,D: $i] :
            ( ! [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ F ) ) ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ F ) ) ) ) )
               => ( ( s @ num @ E )
                  = ( s @ num @ F ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) )
            & ( ( s @ ( fun @ A @ bool ) @ B )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ D ) ) ) @ ( s @ ( fun @ num @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1080]) ).

thf(275,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ C ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumseg) ).

thf(4804,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ C ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ num @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ C ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ num @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(1283,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ D ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ C ) ) @ sndcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) )
          = ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( cart @ A @ C ) ) @ lambda ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ D ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asndcart) ).

thf(10720,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ D ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ C ) ) @ sndcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) )
          = ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( cart @ A @ C ) ) @ lambda ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ D ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1283]) ).

thf(18,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        <=> ? [C: $i,D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ y1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_add) ).

thf(3242,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
           => ? [C: $i,D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ y1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) )
          & ( ? [C: $i,D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ y1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(886,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ C )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_MULu_ABSORB) ).

thf(8458,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ C )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[886]) ).

thf(1615,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ real @ B )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SOSu_EQu_0) ).

thf(12955,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      & ( ( ( ( s @ real @ A )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1615]) ).

thf(1429,axiom,
    ! [A: $i] :
      ( ( ( s @ bool @ A )
        = ( s @ bool @ t ) )
      | ( ( s @ bool @ A )
        = ( s @ bool @ f0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBOOLu_CASESu_AX) ).

thf(11828,plain,
    ! [A: $i] :
      ( ( ( s @ bool @ A )
        = ( s @ bool @ t ) )
      | ( ( s @ bool @ A )
        = ( s @ bool @ f0 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1429]) ).

thf(1354,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i,I: $i,J: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ D @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) ) @ E ) @ ( s @ ( fun @ D @ num ) @ F ) ) ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ F ) @ ( s @ D @ J ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ G ) @ ( s @ D @ J ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ D @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i,J: $i] :
              ( ( s @ ( fun @ D @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ D @ num ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
              = ( s @ ( fun @ D @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) ) @ E ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ D @ num ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) )
         => ! [G: $i] :
              ( ! [H: $i,I: $i,J: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ real ) @ J ) ) ) )
                <=> ! [K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ geq ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ real ) @ J ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ D @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ L ) ) ) @ ( s @ num @ K ) ) ) ) ) ) )
             => ! [H: $i] :
                  ( ! [I: $i,J: $i] :
                      ( ( s @ ( fun @ ( prod @ num @ D ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
                      = ( s @ ( fun @ ( prod @ num @ D ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) ) ) )
                 => ! [I: $i] :
                      ( ! [J: $i,K: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) ) @ I ) @ ( s @ ( fun @ D @ A ) @ J ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) ) ) )
                        <=> ! [L: $i,M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ J ) @ ( s @ D @ M ) ) ) ) ) ) )
                     => ! [J: $i] :
                          ( ! [K: $i,L: $i,M: $i,N: $i,O: $i] :
                              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ D @ num ) @ K ) ) ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) ) ) )
                            <=> ! [P: $i,Q: $i] :
                                ? [R: $i] :
                                  ( ( ( p @ ( s @ bool @ R ) )
                                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ K ) @ ( s @ D @ Q ) ) ) ) ) @ ( s @ num @ P ) ) ) )
                                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ P ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ L ) @ ( s @ D @ Q ) ) ) ) ) )
                                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ Q ) ) ) ) ) )
                                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ P ) ) ) @ ( s @ D @ Q ) ) ) ) ) ) ) @ ( s @ bool @ R ) ) ) ) ) )
                         => ! [K: $i] :
                              ( ! [L: $i,M: $i,N: $i,O: $i] :
                                  ( ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ D @ num ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) )
                                  = ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ D @ num ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) ) ) ) )
                             => ! [L: $i,M: $i,N: $i,O: $i,P: $i,Q: $i] :
                                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ D @ num ) @ P ) ) ) @ ( s @ ( fun @ D @ num ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) ) @ I ) @ ( s @ ( fun @ D @ A ) @ N ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ O ) ) ) ) ) )
                                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ D @ A ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ num ) @ P ) ) ) @ ( s @ ( fun @ D @ num ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ O ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_SUM) ).

thf(11269,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i,I: $i,J: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ D @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) ) @ E ) @ ( s @ ( fun @ D @ num ) @ F ) ) ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ F ) @ ( s @ D @ J ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ G ) @ ( s @ D @ J ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ D @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i,J: $i] :
              ( ( s @ ( fun @ D @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ D @ num ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
              = ( s @ ( fun @ D @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) ) @ E ) @ ( s @ ( fun @ D @ num ) @ G ) ) ) @ ( s @ ( fun @ D @ num ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) )
         => ! [G: $i] :
              ( ! [H: $i,I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ real ) @ J ) ) ) )
                   => ! [K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ geq ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ real ) @ J ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ D @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ L ) ) ) @ ( s @ num @ K ) ) ) ) ) ) )
                  & ( ! [K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ geq ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ real ) @ J ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ D @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ L ) ) ) @ ( s @ num @ K ) ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ real ) @ J ) ) ) ) ) )
             => ! [H: $i] :
                  ( ! [I: $i,J: $i] :
                      ( ( s @ ( fun @ ( prod @ num @ D ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
                      = ( s @ ( fun @ ( prod @ num @ D ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ real ) @ bool ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) ) ) )
                 => ! [I: $i] :
                      ( ! [J: $i,K: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) ) @ I ) @ ( s @ ( fun @ D @ A ) @ J ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) ) ) )
                           => ! [L: $i,M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ J ) @ ( s @ D @ M ) ) ) ) ) ) )
                          & ( ! [L: $i,M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ J ) @ ( s @ D @ M ) ) ) ) ) )
                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) ) @ I ) @ ( s @ ( fun @ D @ A ) @ J ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ K ) ) ) ) ) )
                     => ! [J: $i] :
                          ( ! [K: $i,L: $i,M: $i,N: $i,O: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ D @ num ) @ K ) ) ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) ) ) )
                               => ! [P: $i,Q: $i] :
                                  ? [R: $i] :
                                    ( ( ( p @ ( s @ bool @ R ) )
                                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ K ) @ ( s @ D @ Q ) ) ) ) ) @ ( s @ num @ P ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ P ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ L ) @ ( s @ D @ Q ) ) ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ Q ) ) ) ) ) )
                                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ K ) @ ( s @ D @ Q ) ) ) ) ) @ ( s @ num @ P ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ P ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ L ) @ ( s @ D @ Q ) ) ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ Q ) ) ) ) )
                                     => ( p @ ( s @ bool @ R ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ P ) ) ) @ ( s @ D @ Q ) ) ) ) ) ) ) @ ( s @ bool @ R ) ) ) ) ) )
                              & ( ! [P: $i,Q: $i] :
                                  ? [R: $i] :
                                    ( ( ( p @ ( s @ bool @ R ) )
                                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ K ) @ ( s @ D @ Q ) ) ) ) ) @ ( s @ num @ P ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ P ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ L ) @ ( s @ D @ Q ) ) ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ Q ) ) ) ) ) )
                                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ K ) @ ( s @ D @ Q ) ) ) ) ) @ ( s @ num @ P ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ P ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ D @ num ) @ L ) @ ( s @ D @ Q ) ) ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ D @ Q ) ) ) ) )
                                     => ( p @ ( s @ bool @ R ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) @ ( s @ ( prod @ num @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ num @ D ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ D @ ( prod @ num @ D ) ) ) @ c_ ) @ ( s @ num @ P ) ) ) @ ( s @ D @ Q ) ) ) ) ) ) ) @ ( s @ bool @ R ) ) ) ) )
                               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ D @ num ) @ K ) ) ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ B @ C ) @ N ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ O ) ) ) ) ) )
                         => ! [K: $i] :
                              ( ! [L: $i,M: $i,N: $i,O: $i] :
                                  ( ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ D @ num ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) )
                                  = ( s @ ( fun @ ( prod @ num @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ bool ) @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ D @ num ) @ L ) ) ) @ ( s @ ( fun @ D @ num ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ B @ C ) @ O ) ) ) ) ) )
                             => ! [L: $i,M: $i,N: $i,O: $i,P: $i,Q: $i] :
                                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ D @ num ) @ P ) ) ) @ ( s @ ( fun @ D @ num ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) ) ) @ ( s @ ( fun @ ( prod @ num @ D ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( fun @ ( prod @ num @ D ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ num @ D ) @ A ) @ bool ) ) @ I ) @ ( s @ ( fun @ D @ A ) @ N ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( prod @ num @ D ) @ real ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ O ) ) ) ) ) )
                                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ D @ A ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ D @ num ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ real ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ num ) @ P ) ) ) @ ( s @ ( fun @ D @ num ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ num @ real ) ) ) @ O ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1354]) ).

thf(2385,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ empty ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_SUP) ).

thf(17360,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ empty ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2385]) ).

thf(1559,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRIANGLEu_LEMMA) ).

thf(12632,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1559]) ).

thf(1237,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_LT2u_REV) ).

thf(10423,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1237]) ).

thf(2799,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
    <=> ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSWAPu_FORALLu_THM) ).

thf(19679,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
       => ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ D ) ) ) ) )
      & ( ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ D ) ) ) )
       => ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2799]) ).

thf(2436,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ I ) ) ) ) )
                = ( s @ B @ I ) ) ) )
        & ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) )
                = ( s @ A @ I ) )
              & ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_EQu_GENERALu_INVERSES) ).

thf(17720,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ I ) ) ) ) )
                = ( s @ B @ I ) ) ) )
        & ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) )
                = ( s @ A @ I ) )
              & ( ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2436]) ).

thf(820,axiom,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ u_0 ) ) )
    = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ODDu_conjunct1) ).

thf(8072,plain,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ u_0 ) ) )
    = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[820]) ).

thf(2198,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_CLAUSESu_conjunct5) ).

thf(16468,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2198]) ).

thf(2082,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) )
    <=> ( ! [E: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) )
            = ( s @ A @ E ) )
        & ! [E: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) )
            = ( s @ B @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aISO) ).

thf(15788,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) )
       => ( ! [E: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) )
              = ( s @ A @ E ) )
          & ! [E: $i] :
              ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) )
              = ( s @ B @ E ) ) ) )
      & ( ( ! [E: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) )
              = ( s @ A @ E ) )
          & ! [E: $i] :
              ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) )
              = ( s @ B @ E ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2082]) ).

thf(2753,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] : ( p @ ( s @ bool @ B ) )
    <=> ( p @ ( s @ bool @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_SIMP) ).

thf(19393,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2753]) ).

thf(105,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_INVu_EQ) ).

thf(3743,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(2463,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_SUBu_COMPONENT) ).

thf(17873,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2463]) ).

thf(1834,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) )
            = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
         => ( ( s @ ( list @ B ) @ D )
            = ( s @ ( list @ B ) @ E ) ) )
    <=> ! [D: $i,E: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
         => ( ( s @ B @ D )
            = ( s @ B @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJECTIVEu_MAP) ).

thf(14392,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i,E: $i] :
            ( ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) )
              = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
           => ( ( s @ ( list @ B ) @ D )
              = ( s @ ( list @ B ) @ E ) ) )
       => ! [D: $i,E: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
           => ( ( s @ B @ D )
              = ( s @ B @ E ) ) ) )
      & ( ! [D: $i,E: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
           => ( ( s @ B @ D )
              = ( s @ B @ E ) ) )
       => ! [D: $i,E: $i] :
            ( ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) )
              = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
           => ( ( s @ ( list @ B ) @ D )
              = ( s @ ( list @ B ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1834]) ).

thf(677,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRealu_exto_REALu_INV2u_conjunct0) ).

thf(7221,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[677]) ).

thf(440,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) )
     => ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCHOOSEu_SUBSETu_BETWEEN) ).

thf(5900,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) )
     => ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).

thf(874,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ADDu_conjunct4) ).

thf(8410,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[874]) ).

thf(3063,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ E ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSPu_COMPONENT) ).

thf(21162,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ E ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3063]) ).

thf(1964,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_CIRCLE) ).

thf(15200,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1964]) ).

thf(2616,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_SYM) ).

thf(18714,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2616]) ).

thf(545,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_2) ).

thf(6475,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).

thf(1693,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_GEu_SQUARE) ).

thf(13483,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1693]) ).

thf(458,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_RNEG) ).

thf(5973,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).

thf(377,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUBu_conjunct1) ).

thf(5498,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).

thf(1853,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ empty ) )
    <=> ( ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
        & ( ( s @ ( fun @ A @ bool ) @ C )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEMPTYu_UNION) ).

thf(14509,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
       => ( ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ empty ) )
          & ( ( s @ ( fun @ A @ bool ) @ C )
            = ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
      & ( ( ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ empty ) )
          & ( ( s @ ( fun @ A @ bool ) @ C )
            = ( s @ ( fun @ A @ bool ) @ empty ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1853]) ).

thf(764,axiom,
    ! [A: $i] :
      ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ B ) ) ) )
    <=> ! [B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n20 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_VECTORu_2) ).

thf(7738,plain,
    ! [A: $i] :
      ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ B ) ) ) )
       => ! [B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n20 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) )
      & ( ! [B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n20 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) )
       => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[764]) ).

thf(2595,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        | ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_INu_CLAUSESu_conjunct1) ).

thf(18608,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          | ? [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          | ? [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) )
       => ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2595]) ).

thf(2897,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ B ) @ outr ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ y ) ) ) ) )
      = ( s @ B @ y ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aOUTR) ).

thf(20229,plain,
    ! [A: $i,B: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ B ) @ outr ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ y ) ) ) ) )
      = ( s @ B @ y ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2897]) ).

thf(36,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
    <=> ? [G: $i] :
          ( ( ( s @ ( fun @ A @ bool ) @ E )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
          & ( ( s @ B @ F )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ G ) ) ) @ ( s @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINRECu_1u_LEMMA) ).

thf(3390,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ? [G: $i] :
            ( ( ( s @ ( fun @ A @ bool ) @ E )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
            & ( ( s @ B @ F )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ G ) ) ) @ ( s @ B @ D ) ) ) ) ) )
      & ( ? [G: $i] :
            ( ( ( s @ ( fun @ A @ bool ) @ E )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
            & ( ( s @ B @ F )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ G ) ) ) @ ( s @ B @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(2549,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_IMPu_SUPERADMISSIBLE) ).

thf(18347,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2549]) ).

thf(14,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ A )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_NZ) ).

thf(3222,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ A )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(1193,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_UNIONu_RZERO) ).

thf(10148,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1193]) ).

thf(1203,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ A ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ B ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_OFu_NUMu_EQ) ).

thf(10207,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ A ) ) )
          = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ B ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ A ) ) )
          = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1203]) ).

thf(149,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ A ) ) ) @ column ) @ ( s @ num @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOLUMNu_TRANSP) ).

thf(3937,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ A ) ) ) @ column ) @ ( s @ num @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(2497,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_RVSUM) ).

thf(18027,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2497]) ).

thf(2800,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_LADD) ).

thf(19684,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2800]) ).

thf(3056,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) )
    <=> ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_PAIRu_THM) ).

thf(21119,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) )
       => ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) ) )
      & ( ! [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) )
       => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3056]) ).

thf(3060,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_INFINITE) ).

thf(21146,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3060]) ).

thf(1981,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
    <=> ~ ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_NOTu_THM) ).

thf(15285,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ~ ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
      & ( ~ ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ? [C: $i] :
            ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1981]) ).

thf(2984,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_REFL) ).

thf(20681,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2984]) ).

thf(1356,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_ADDR) ).

thf(11315,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1356]) ).

thf(1593,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ) ) @ ( s @ real @ z ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ y ) ) ) @ ( s @ real @ z ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_ACIu_conjunct1) ).

thf(12838,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ) ) @ ( s @ real @ z ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ y ) ) ) @ ( s @ real @ z ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1593]) ).

thf(1994,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ t ) ) ) @ ( s @ A @ B ) ) ) @ ( s @ A @ C ) ) )
        = ( s @ A @ B ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ f0 ) ) ) @ ( s @ A @ B ) ) ) @ ( s @ A @ C ) ) )
        = ( s @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONDu_CLAUSES) ).

thf(15399,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ t ) ) ) @ ( s @ A @ B ) ) ) @ ( s @ A @ C ) ) )
        = ( s @ A @ B ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ f0 ) ) ) @ ( s @ A @ B ) ) ) @ ( s @ A @ C ) ) )
        = ( s @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1994]) ).

thf(1175,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ A )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_EQ) ).

thf(10047,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ A )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1175]) ).

thf(838,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_MULT) ).

thf(8191,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[838]) ).

thf(1858,axiom,
    ! [A: $i] :
      ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) )
        & ! [B: $i,C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
        & ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSITIVEu_STEPWISEu_LE) ).

thf(14526,plain,
    ! [A: $i] :
      ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) )
        & ! [B: $i,C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
        & ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1858]) ).

thf(1254,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLENGTHu_LISTu_OFu_SET) ).

thf(10533,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1254]) ).

thf(2127,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_REALu_INTERVALu_conjunct7) ).

thf(16058,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2127]) ).

thf(81,axiom,
    ! [A: $i] :
      ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ B ) ) ) )
    <=> ? [B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n20 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_VECTORu_2) ).

thf(3628,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ B ) ) ) )
       => ? [B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n20 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) ) )
      & ( ? [B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n20 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n20 ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(615,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ADDu_conjunct8) ).

thf(6908,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).

thf(885,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_LMULu_EQ) ).

thf(8455,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[885]) ).

thf(2156,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ( ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
        | ? [C: $i,D: $i] :
            ( ( ( s @ ( fun @ A @ bool ) @ B )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_CASES) ).

thf(16237,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ( ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ empty ) )
          | ? [C: $i,D: $i] :
              ( ( ( s @ ( fun @ A @ bool ) @ B )
                = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
      & ( ( ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ empty ) )
          | ? [C: $i,D: $i] :
              ( ( ( s @ ( fun @ A @ bool ) @ B )
                = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2156]) ).

thf(2664,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOIDALu_ADD) ).

thf(18934,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2664]) ).

thf(158,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_MULu_EQu_conjunct1) ).

thf(3967,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(1219,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
        <=> ? [G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ? [I: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ B @ G )
                      = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ I ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGE) ).

thf(10302,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
           => ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( ( s @ B @ G )
                        = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ I ) ) ) ) ) )
                & ( ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( ( s @ B @ G )
                        = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ I ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
          & ( ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( ( s @ B @ G )
                        = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ I ) ) ) ) ) )
                & ( ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( ( s @ B @ G )
                        = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ I ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1219]) ).

thf(1545,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ C ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_BASIS) ).

thf(12533,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ C ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1545]) ).

thf(2637,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_MAX) ).

thf(18802,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2637]) ).

thf(2987,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_ASSOC) ).

thf(20698,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2987]) ).

thf(2646,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ A ) @ ( s @ ind0 @ indu_0 ) ) ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ A ) @ ( s @ ind0 @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ A ) @ ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ B ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ A ) @ ( s @ ind0 @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMu_REPu_INDUCT) ).

thf(18845,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ A ) @ ( s @ ind0 @ indu_0 ) ) ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ A ) @ ( s @ ind0 @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ A ) @ ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ B ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ A ) @ ( s @ ind0 @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2646]) ).

thf(825,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) ) ) )
        <=> ! [G: $i,H: $i] :
            ? [I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ! [J: $i] :
                    ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ E ) @ ( s @ B @ F ) ) )
              = ( s @ bool @ t ) )
         => ! [F: $i] :
              ( ! [G: $i] :
                  ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ F ) @ ( s @ ( fun @ A @ C ) @ G ) ) )
                  = ( s @ ( fun @ B @ bool ) @ E ) )
             => ! [G: $i] :
                  ( ! [H: $i,I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ G ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ I ) ) ) )
                    <=> ! [J: $i,K: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ I ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ K ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ K ) ) ) ) ) ) )
                 => ! [H: $i] :
                      ( ( ? [I: $i] :
                            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ I ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ G ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ I ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ H ) ) ) ) )
                        & ! [I: $i,J: $i,K: $i,L: $i,M: $i,N: $i,O: $i] :
                            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ H ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ L ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ H ) ) ) ) )
                           => ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) )
                                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ K ) @ ( s @ B @ O ) ) ) )
                             => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ M ) ) ) @ ( s @ B @ N ) ) )
                                = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ L ) @ ( s @ ( fun @ A @ C ) @ M ) ) ) @ ( s @ B @ O ) ) ) ) ) ) )
                     => ? [I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ H ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRECURSIONu_CASEWISE) ).

thf(8090,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) ) ) )
           => ! [G: $i,H: $i] :
              ? [I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) ) )
                & ( ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
          & ( ! [G: $i,H: $i] :
              ? [I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) ) )
                & ( ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ E ) @ ( s @ B @ F ) ) )
              = ( s @ bool @ t ) )
         => ! [F: $i] :
              ( ! [G: $i] :
                  ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ F ) @ ( s @ ( fun @ A @ C ) @ G ) ) )
                  = ( s @ ( fun @ B @ bool ) @ E ) )
             => ! [G: $i] :
                  ( ! [H: $i,I: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ G ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ I ) ) ) )
                       => ! [J: $i,K: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ I ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ K ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ K ) ) ) ) ) ) )
                      & ( ! [J: $i,K: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ I ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ K ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ K ) ) ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ G ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ I ) ) ) ) ) )
                 => ! [H: $i] :
                      ( ( ? [I: $i] :
                            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ I ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ G ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ I ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ H ) ) ) ) )
                        & ! [I: $i,J: $i,K: $i,L: $i,M: $i,N: $i,O: $i] :
                            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ H ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ L ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ H ) ) ) ) )
                           => ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) )
                                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ K ) @ ( s @ B @ O ) ) ) )
                             => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ M ) ) ) @ ( s @ B @ N ) ) )
                                = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ L ) @ ( s @ ( fun @ A @ C ) @ M ) ) ) @ ( s @ B @ O ) ) ) ) ) ) )
                     => ? [I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ H ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[825]) ).

thf(421,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
          <=> ( ( s @ num @ D )
              = ( s @ num @ E ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ E ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATu_COMPONENT) ).

thf(5801,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
           => ( ( s @ num @ D )
              = ( s @ num @ E ) ) )
          & ( ( ( s @ num @ D )
              = ( s @ num @ E ) )
           => ( p @ ( s @ bool @ F ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ E ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).

thf(2367,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        | ( ( s @ A @ D )
          = ( s @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_DEF) ).

thf(17267,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          | ( ( s @ A @ D )
            = ( s @ A @ C ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          | ( ( s @ A @ D )
            = ( s @ A @ C ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2367]) ).

thf(2452,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_NUM) ).

thf(17823,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2452]) ).

thf(1591,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_LID) ).

thf(12800,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1591]) ).

thf(1199,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ bool ) ) @ A ) @ ( s @ int @ B ) ) ) @ ( s @ num @ C ) ) ) )
        <=> ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ C ) ) )
            = ( s @ int @ B ) ) )
     => ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ int @ num ) @ numu_ofu_int ) @ ( s @ int @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ h_ ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ bool ) ) @ A ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_ofu_int) ).

thf(10180,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ bool ) ) @ A ) @ ( s @ int @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ C ) ) )
              = ( s @ int @ B ) ) )
          & ( ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ C ) ) )
              = ( s @ int @ B ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ bool ) ) @ A ) @ ( s @ int @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ int @ num ) @ numu_ofu_int ) @ ( s @ int @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ h_ ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ bool ) ) @ A ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1199]) ).

thf(1783,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ B @ C ) @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( cart @ B @ C ) ) @ sndcart ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ D ) ) ) @ ( s @ ( cart @ B @ C ) @ E ) ) ) ) )
      = ( s @ ( cart @ B @ C ) @ E ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSNDCARTu_PASTECART) ).

thf(14116,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ B @ C ) @ ( i @ ( s @ ( fun @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( cart @ B @ C ) ) @ sndcart ) @ ( s @ ( cart @ B @ ( finite_sum @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( cart @ B @ C ) @ ( cart @ B @ ( finite_sum @ A @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ B @ A ) @ D ) ) ) @ ( s @ ( cart @ B @ C ) @ E ) ) ) ) )
      = ( s @ ( cart @ B @ C ) @ E ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1783]) ).

thf(2589,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ C ) )
        & ( ( s @ num @ B )
          = ( s @ num @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMPAIRu_INJ) ).

thf(18581,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ C ) )
          & ( ( s @ num @ B )
            = ( s @ num @ D ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ C ) )
          & ( ( s @ num @ B )
            = ( s @ num @ D ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2589]) ).

thf(775,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LEu_conjunct8) ).

thf(7826,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[775]) ).

thf(176,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_NEGTOTAL) ).

thf(4069,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(1667,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ D ) ) ) ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_POSu_EQu_0u_NUMSEG) ).

thf(13340,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ D ) ) ) ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1667]) ).

thf(219,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUBu_conjunct7) ).

thf(4416,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(1694,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_BOUND) ).

thf(13490,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1694]) ).

thf(1448,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_LCANCELu_IMP) ).

thf(11927,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1448]) ).

thf(701,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_SUBu_LADD) ).

thf(7333,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[701]) ).

thf(2101,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ onto ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) )
    <=> ! [D: $i] :
        ? [E: $i] :
          ( ( s @ A @ D )
          = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aONTO) ).

thf(15934,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ onto ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) )
       => ! [D: $i] :
          ? [E: $i] :
            ( ( s @ A @ D )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) )
      & ( ! [D: $i] :
          ? [E: $i] :
            ( ( s @ A @ D )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ onto ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2101]) ).

thf(2680,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_UNION) ).

thf(18998,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2680]) ).

thf(3085,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_ADD) ).

thf(21270,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3085]) ).

thf(635,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_RMUL) ).

thf(6988,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[635]) ).

thf(1152,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_SUB) ).

thf(9942,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1152]) ).

thf(1828,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_TOTAL) ).

thf(14352,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1828]) ).

thf(1322,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SGNu_EQu_conjunct1) ).

thf(11000,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1322]) ).

thf(1438,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_INDUCTu_STRONG) ).

thf(11873,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1438]) ).

thf(7,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ B ) @ num ) @ length ) @ ( s @ ( list @ B ) @ E ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( list @ C ) @ num ) @ length ) @ ( s @ ( list @ C ) @ F ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) ) @ map2 ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) @ ( s @ ( list @ C ) @ F ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( list @ C ) @ num ) @ length ) @ ( s @ ( list @ C ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLENGTHu_MAP2) ).

thf(3201,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ B ) @ num ) @ length ) @ ( s @ ( list @ B ) @ E ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( list @ C ) @ num ) @ length ) @ ( s @ ( list @ C ) @ F ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) ) @ map2 ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) @ ( s @ ( list @ C ) @ F ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( list @ C ) @ num ) @ length ) @ ( s @ ( list @ C ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(2276,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) )
      = ( s @ ( cart @ real @ A ) @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_SUBu_ADD) ).

thf(16854,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) )
      = ( s @ ( cart @ real @ A ) @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2276]) ).

thf(209,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LT2) ).

thf(4361,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(1493,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) ) ) )
        <=> ! [G: $i,H: $i] :
            ? [I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ! [J: $i] :
                    ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) )
            <=> ! [I: $i,J: $i] :
                ? [K: $i] :
                  ( ( ( p @ ( s @ bool @ K ) )
                  <=> ! [L: $i,M: $i,N: $i] :
                        ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ M ) ) )
                          = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                       => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                          = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ F ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) ) ) )
                <=> ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) ) )
             => ! [G: $i] :
                  ( ! [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ G ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) )
                    <=> ! [I: $i,J: $i] :
                        ? [K: $i] :
                          ( ( ( p @ ( s @ bool @ K ) )
                          <=> ! [L: $i,M: $i,N: $i] :
                                ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ M ) ) )
                                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                               => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                                  = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
                 => ! [H: $i] :
                      ( ! [I: $i] :
                          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ H ) @ ( s @ B @ I ) ) )
                          = ( s @ bool @ t ) )
                     => ! [I: $i] :
                          ( ! [J: $i] :
                              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) @ ( s @ ( fun @ A @ C ) @ J ) ) )
                              = ( s @ ( fun @ B @ bool ) @ H ) )
                         => ! [J: $i] :
                              ( ! [K: $i,L: $i] :
                                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ J ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) ) ) )
                                <=> ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) ) )
                             => ! [K: $i] :
                                  ( ( ? [L: $i] :
                                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ L ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ J ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ L ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ G ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ F ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) )
                                 => ? [L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRECURSIONu_SUPERADMISSIBLE) ).

thf(12138,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) ) ) )
           => ! [G: $i,H: $i] :
              ? [I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) ) )
                & ( ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
          & ( ! [G: $i,H: $i] :
              ? [I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) ) )
                & ( ! [J: $i] :
                      ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ J ) ) ) ) )
                      = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ B @ J ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ F ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) )
               => ! [I: $i,J: $i] :
                  ? [K: $i] :
                    ( ( ( p @ ( s @ bool @ K ) )
                     => ! [L: $i,M: $i,N: $i] :
                          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ M ) ) )
                            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                         => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                            = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                    & ( ! [L: $i,M: $i,N: $i] :
                          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ M ) ) )
                            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                         => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                            = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) )
                     => ( p @ ( s @ bool @ K ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
              & ( ! [I: $i,J: $i] :
                  ? [K: $i] :
                    ( ( ( p @ ( s @ bool @ K ) )
                     => ! [L: $i,M: $i,N: $i] :
                          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ M ) ) )
                            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                         => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                            = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                    & ( ! [L: $i,M: $i,N: $i] :
                          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ M ) ) )
                            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                         => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                            = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) )
                     => ( p @ ( s @ bool @ K ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ F ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) ) ) )
                   => ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) ) )
                  & ( ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ F ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ G ) ) ) ) ) )
             => ! [G: $i] :
                  ( ! [H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ G ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) )
                       => ! [I: $i,J: $i] :
                          ? [K: $i] :
                            ( ( ( p @ ( s @ bool @ K ) )
                             => ! [L: $i,M: $i,N: $i] :
                                  ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ M ) ) )
                                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                                 => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                            & ( ! [L: $i,M: $i,N: $i] :
                                  ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ M ) ) )
                                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                                 => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) )
                             => ( p @ ( s @ bool @ K ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
                      & ( ! [I: $i,J: $i] :
                          ? [K: $i] :
                            ( ( ( p @ ( s @ bool @ K ) )
                             => ! [L: $i,M: $i,N: $i] :
                                  ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ M ) ) )
                                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                                 => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) ) )
                            & ( ! [L: $i,M: $i,N: $i] :
                                  ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ M ) ) )
                                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ I ) @ ( s @ B @ N ) ) ) )
                                 => ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ M ) ) )
                                    = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) @ ( s @ B @ N ) ) ) ) )
                             => ( p @ ( s @ bool @ K ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ G ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ H ) ) ) ) ) )
                 => ! [H: $i] :
                      ( ! [I: $i] :
                          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ H ) @ ( s @ B @ I ) ) )
                          = ( s @ bool @ t ) )
                     => ! [I: $i] :
                          ( ! [J: $i] :
                              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) @ ( s @ ( fun @ A @ C ) @ J ) ) )
                              = ( s @ ( fun @ B @ bool ) @ H ) )
                         => ! [J: $i] :
                              ( ! [K: $i,L: $i] :
                                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ J ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) ) ) )
                                   => ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) ) )
                                  & ( ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) @ ( s @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ B @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ N ) ) ) ) ) )
                                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ J ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ L ) ) ) ) ) )
                             => ! [K: $i] :
                                  ( ( ? [L: $i] :
                                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ L ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ J ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ L ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ G ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ bool ) @ F ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) )
                                 => ? [L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ C ) @ L ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) ) ) @ K ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1493]) ).

thf(50,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_0) ).

thf(3454,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(122,axiom,
    ! [A: $i] :
      ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EXISTSu_ABS) ).

thf(3802,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) )
      & ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(433,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_3) ).

thf(5872,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).

thf(1912,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ? [E: $i] :
          ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
          & ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ B @ F )
                = ( s @ B @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aleu_c) ).

thf(14848,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ? [E: $i] :
            ( ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
            & ! [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
               => ( ( s @ B @ F )
                  = ( s @ B @ G ) ) ) ) )
      & ( ? [E: $i] :
            ( ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
            & ! [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
               => ( ( s @ B @ F )
                  = ( s @ B @ G ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1912]) ).

thf(1765,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ? [C: $i,D: $i] :
          ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVMODu_EXIST) ).

thf(13932,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ? [C: $i,D: $i] :
          ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1765]) ).

thf(671,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ union ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_ADDu_SPLIT) ).

thf(7193,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ union ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[671]) ).

thf(1329,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ C @ B ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ C @ B ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ C @ B ) @ D ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPPORTu_CLAUSESu_conjunct6) ).

thf(11031,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ C @ B ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ C @ B ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ C @ B ) @ D ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1329]) ).

thf(2705,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) ) @ F ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ H ) ) ) )
        <=> ! [I: $i,J: $i,K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ geq ) @ ( s @ E @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ H ) @ ( s @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ ( prod @ B @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ c_ ) @ ( s @ B @ J ) ) ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) ) ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ G ) @ ( s @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ ( prod @ B @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ c_ ) @ ( s @ B @ J ) ) ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) ) ) ) ) ) )
     => ! [G: $i] :
          ( ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) ) @ F ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ G ) ) ) ) )
          = ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ G ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIREDu_ETAu_THMu_conjunct2) ).

thf(19149,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) ) @ F ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ H ) ) ) )
           => ! [I: $i,J: $i,K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ geq ) @ ( s @ E @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ H ) @ ( s @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ ( prod @ B @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ c_ ) @ ( s @ B @ J ) ) ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) ) ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ G ) @ ( s @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ ( prod @ B @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ c_ ) @ ( s @ B @ J ) ) ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) ) ) ) ) ) )
          & ( ! [I: $i,J: $i,K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ geq ) @ ( s @ E @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ H ) @ ( s @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ ( prod @ B @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ c_ ) @ ( s @ B @ J ) ) ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) ) ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ G ) @ ( s @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ ( prod @ B @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ c_ ) @ ( s @ B @ J ) ) ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ K ) ) ) @ ( s @ D @ L ) ) ) ) ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) ) @ F ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ H ) ) ) ) ) )
     => ! [G: $i] :
          ( ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ bool ) ) @ F ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ G ) ) ) ) )
          = ( s @ ( fun @ ( prod @ A @ ( prod @ B @ ( prod @ C @ D ) ) ) @ E ) @ G ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2705]) ).

thf(1682,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) )
        <=> ? [I: $i,J: $i] :
              ( ( ( p @ ( s @ bool @ J ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                    = ( s @ A @ G ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ H ) ) ) @ ( s @ A @ I ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ A @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ H ) ) ) )
         => ! [F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
             => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_IMAGEu_GEN) ).

thf(13398,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) )
           => ? [I: $i,J: $i] :
                ( ( ( p @ ( s @ bool @ J ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                      = ( s @ A @ G ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                      = ( s @ A @ G ) ) )
                 => ( p @ ( s @ bool @ J ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) ) )
          & ( ? [I: $i,J: $i] :
                ( ( ( p @ ( s @ bool @ J ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                      = ( s @ A @ G ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                      = ( s @ A @ G ) ) )
                 => ( p @ ( s @ bool @ J ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ H ) ) ) @ ( s @ A @ I ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ A @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ H ) ) ) )
         => ! [F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
             => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1682]) ).

thf(978,axiom,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_conjunct0) ).

thf(8971,plain,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[978]) ).

thf(2602,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_ANTISYM) ).

thf(18640,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2602]) ).

thf(141,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_DOUBLE) ).

thf(3903,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(2554,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_SYM) ).

thf(18364,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2554]) ).

thf(1982,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPSUBSET) ).

thf(15290,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( ( s @ ( fun @ A @ bool ) @ B )
           != ( s @ ( fun @ A @ bool ) @ C ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( ( s @ ( fun @ A @ bool ) @ B )
           != ( s @ ( fun @ A @ bool ) @ C ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1982]) ).

thf(2917,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_ADD2u_ABSORBu_LT) ).

thf(20303,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2917]) ).

thf(208,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
      <=> ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_SUPu_FINITE) ).

thf(4352,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
         => ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
        & ( ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(560,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ min ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_MIN) ).

thf(6533,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ min ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).

thf(2115,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_MUL) ).

thf(15991,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2115]) ).

thf(1584,axiom,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ l0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aELu_conjunct0) ).

thf(12751,plain,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ l0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1584]) ).

thf(248,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
      <=> ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
           => ( ( s @ B @ E )
              = ( s @ B @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_IMAGEu_EQu_INJ) ).

thf(4644,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
             => ( ( s @ B @ E )
                = ( s @ B @ F ) ) ) )
        & ( ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
             => ( ( s @ B @ E )
                = ( s @ B @ F ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(1933,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ? [D: $i] :
          ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPSUBSETu_INSERTu_SUBSET) ).

thf(15042,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ? [D: $i] :
            ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1933]) ).

thf(2374,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
        & ! [G: $i,H: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
              & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ D ) @ ( s @ B @ G ) ) )
                = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ D ) @ ( s @ B @ H ) ) ) ) )
           => ( ( s @ B @ G )
              = ( s @ B @ H ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ C ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ C @ ( cart @ real @ A ) ) @ E ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ C @ ( cart @ real @ A ) ) @ E ) ) ) @ ( s @ ( fun @ B @ C ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_IMAGE) ).

thf(17293,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
        & ! [G: $i,H: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
              & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ D ) @ ( s @ B @ G ) ) )
                = ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ D ) @ ( s @ B @ H ) ) ) ) )
           => ( ( s @ B @ G )
              = ( s @ B @ H ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ C @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ C ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ C @ ( cart @ real @ A ) ) @ E ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ C @ ( cart @ real @ A ) ) @ E ) ) ) @ ( s @ ( fun @ B @ C ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2374]) ).

thf(1909,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_LEu_TRANS) ).

thf(14821,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1909]) ).

thf(2541,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_SING) ).

thf(18308,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2541]) ).

thf(1646,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_LMULu_EQ) ).

thf(13213,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1646]) ).

thf(1997,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
        = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_RZERO) ).

thf(15414,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
        = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1997]) ).

thf(1509,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_POSu_LT) ).

thf(12282,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1509]) ).

thf(931,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
     => ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ? [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) )
                  = ( s @ A @ D ) ) ) )
      <=> ! [D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ D ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) )
           => ( ( s @ A @ D )
              = ( s @ A @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJECTIVEu_IFFu_INJECTIVE) ).

thf(8739,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
     => ( ( ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
             => ? [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) )
                    = ( s @ A @ D ) ) ) )
         => ! [D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ D ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) )
             => ( ( s @ A @ D )
                = ( s @ A @ E ) ) ) )
        & ( ! [D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ D ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) )
             => ( ( s @ A @ D )
                = ( s @ A @ E ) ) )
         => ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
             => ? [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) )
                    = ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[931]) ).

thf(196,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_CMULu_LID) ).

thf(4288,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(2401,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDELETEu_SUBSET) ).

thf(17499,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2401]) ).

thf(2522,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
        = ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ empty ) )
    <=> ( ( ( s @ ( fun @ A @ bool ) @ C )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
        | ( ( s @ ( fun @ B @ bool ) @ D )
          = ( s @ ( fun @ B @ bool ) @ empty ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCROSSu_EQu_EMPTY) ).

thf(18159,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
          = ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ empty ) )
       => ( ( ( s @ ( fun @ A @ bool ) @ C )
            = ( s @ ( fun @ A @ bool ) @ empty ) )
          | ( ( s @ ( fun @ B @ bool ) @ D )
            = ( s @ ( fun @ B @ bool ) @ empty ) ) ) )
      & ( ( ( ( s @ ( fun @ A @ bool ) @ C )
            = ( s @ ( fun @ A @ bool ) @ empty ) )
          | ( ( s @ ( fun @ B @ bool ) @ D )
            = ( s @ ( fun @ B @ bool ) @ empty ) ) )
       => ( ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
          = ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2522]) ).

thf(712,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_MUL) ).

thf(7407,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[712]) ).

thf(2397,axiom,
    ! [A: $i] :
      ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
    <=> ? [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
             => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_WOP) ).

thf(17482,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
       => ? [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
            & ! [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
               => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ C ) ) ) ) ) ) )
      & ( ? [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
            & ! [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
               => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ C ) ) ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2397]) ).

thf(899,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ A @ E )
           != ( s @ A @ F ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ E ) ) ) ) ) )
         => ! [H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
             => ? [I: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ B @ I ) ) ) @ ( s @ num @ E ) ) ) )
                  & ( ( s @ B @ G )
                    = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINRECu_SUCu_LEMMA) ).

thf(8512,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ A @ E )
           != ( s @ A @ F ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ E ) ) ) ) ) )
         => ! [H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
             => ? [I: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ B @ I ) ) ) @ ( s @ num @ E ) ) ) )
                  & ( ( s @ B @ G )
                    = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[899]) ).

thf(2012,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ B @ E ) ) ) )
        <=> ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ E ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ E ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ C ) @ ( s @ B @ E ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) )
             => ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ f ) ) ) @ ( s @ ( list @ B ) @ E ) ) )
                = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ f ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_EQu_ALL2) ).

thf(15499,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ B @ E ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ E ) ) ) ) )
          & ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) @ ( s @ B @ E ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ C ) @ ( s @ B @ E ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ B ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) )
             => ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ f ) ) ) @ ( s @ ( list @ B ) @ E ) ) )
                = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ f ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2012]) ).

thf(2035,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) )
      = ( s @ ( fun @ A @ bool ) @ univ ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_UNIVu_conjunct1) ).

thf(15592,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) )
      = ( s @ ( fun @ A @ bool ) @ univ ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2035]) ).

thf(2969,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
      <=> ? [D: $i] :
            ( ( s @ ( fun @ A @ bool ) @ C )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aISu_HULL) ).

thf(20589,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ? [D: $i] :
              ( ( s @ ( fun @ A @ bool ) @ C )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        & ( ? [D: $i] :
              ( ( s @ ( fun @ A @ bool ) @ C )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2969]) ).

thf(3,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ num @ B )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_EQu_0) ).

thf(3183,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(1604,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      <=> ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_UNIONu_OVERLAPu_EQ) ).

thf(12889,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
         => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
            = ( s @ ( fun @ A @ bool ) @ empty ) ) )
        & ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
            = ( s @ ( fun @ A @ bool ) @ empty ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1604]) ).

thf(2450,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ C @ B ) @ E ) ) ) @ ( s @ C @ F ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ E ) @ ( s @ C @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aou_THM) ).

thf(17812,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ C @ B ) @ E ) ) ) @ ( s @ C @ F ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ E ) @ ( s @ C @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2450]) ).

thf(2901,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_POSET) ).

thf(20241,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2901]) ).

thf(3124,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_DIFF) ).

thf(21485,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3124]) ).

thf(3117,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_RMUL) ).

thf(21441,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3117]) ).

thf(997,axiom,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
      <=> ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ empty ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_CLAUSESu_conjunct0) ).

thf(9044,plain,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
       => ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ B ) ) )
      & ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ empty ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[997]) ).

thf(2462,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
    <=> ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_INu_UNIONS) ).

thf(17863,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
       => ! [D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) )
      & ( ! [D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2462]) ).

thf(2152,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( ( s @ A @ D )
            = ( s @ A @ C ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNWINDu_THM2) ).

thf(16220,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] :
            ( ( ( s @ A @ D )
              = ( s @ A @ C ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ? [D: $i] :
            ( ( ( s @ A @ D )
              = ( s @ A @ C ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2152]) ).

thf(2715,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_MUL2) ).

thf(19198,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2715]) ).

thf(1275,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ num ) @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ num @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( prod @ num @ num ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( prod @ num @ num ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_SQUAREu_NUM) ).

thf(10656,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ num @ num ) @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ num @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( prod @ num @ num ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( prod @ num @ num ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1275]) ).

thf(2239,axiom,
    ! [A: $i] :
      ( ( ! [B: $i,C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ B ) ) ) @ ( s @ A @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_l_ ) @ ( s @ A @ B ) ) ) @ ( s @ A @ C ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_l_ ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_SUBSET) ).

thf(16666,plain,
    ! [A: $i] :
      ( ( ! [B: $i,C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ B ) ) ) @ ( s @ A @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_l_ ) @ ( s @ A @ B ) ) ) @ ( s @ A @ C ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_l_ ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2239]) ).

thf(984,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_CMULu_LZERO) ).

thf(9008,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[984]) ).

thf(2020,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ A )
        = ( s @ int @ B ) )
    <=> ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_eq) ).

thf(15535,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ A )
          = ( s @ int @ B ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) )
      & ( ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) )
       => ( ( s @ int @ A )
          = ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2020]) ).

thf(2719,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_COMM) ).

thf(19227,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2719]) ).

thf(2041,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_TOTAL) ).

thf(15624,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2041]) ).

thf(710,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LNEGu_UNIQ) ).

thf(7397,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[710]) ).

thf(2792,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( ( p @ ( s @ bool @ E ) )
      <=> ( ( s @ A @ B )
          = ( s @ A @ C ) ) )
      & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ A @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDELETEu_INSERT) ).

thf(19630,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( ( p @ ( s @ bool @ E ) )
       => ( ( s @ A @ B )
          = ( s @ A @ C ) ) )
      & ( ( ( s @ A @ B )
          = ( s @ A @ C ) )
       => ( p @ ( s @ bool @ E ) ) )
      & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ A @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2792]) ).

thf(2869,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
        <=> ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
        <=> ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ C ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ F ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_CURRY) ).

thf(20096,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
           => ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
          & ( ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) ) ) )
     => ! [E: $i] :
          ( ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
           => ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ C ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ F ) ) ) ) ) ) ) ) )
          & ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ C ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ F ) ) ) ) ) ) ) )
           => ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2869]) ).

thf(1464,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ( ! [D: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
        & ! [D: $i,E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_CLAUSESu_GEN) ).

thf(11987,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ( ! [D: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
        & ! [D: $i,E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1464]) ).

thf(855,axiom,
    ! [A: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_EQu_0) ).

thf(8279,plain,
    ! [A: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[855]) ).

thf(2211,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i,E: $i] :
          ( ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C )
          = ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
    <=> ( ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ repu_prod ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( prod @ A @ B ) ) @ absu_prod ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) ) ) ) )
        = ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aprodu_tybiju_conjunct1) ).

thf(16528,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i,E: $i] :
            ( ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C )
            = ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
       => ( ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ repu_prod ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( prod @ A @ B ) ) @ absu_prod ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) ) ) ) )
          = ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) ) )
      & ( ( ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ repu_prod ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( prod @ A @ B ) ) @ absu_prod ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) ) ) ) )
          = ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) )
       => ? [D: $i,E: $i] :
            ( ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C )
            = ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2211]) ).

thf(2487,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_CMULu_NUMSEG) ).

thf(17984,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2487]) ).

thf(2443,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
        & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) )
         != ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_SUPPORT) ).

thf(17767,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) )
           != ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) )
           != ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2443]) ).

thf(2232,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_LEu_REFL) ).

thf(16644,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2232]) ).

thf(2125,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_SUBSET) ).

thf(16048,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2125]) ).

thf(1955,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
         != ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) )
     => ? [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSEGu_PROPERu_SUBSET) ).

thf(15136,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B )
         != ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) )
     => ? [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1955]) ).

thf(2240,axiom,
    ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNCOUNTABLEu_REAL) ).

thf(16669,plain,
    ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2240]) ).

thf(2172,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSPu_MATRIXu_ADD) ).

thf(16329,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2172]) ).

thf(1642,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ B ) @ snd ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) )
      = ( s @ B @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSND) ).

thf(13196,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ B ) @ snd ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) )
      = ( s @ B @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1642]) ).

thf(972,axiom,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ x ) ) ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ x ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ahrealu_invu_th) ).

thf(8955,plain,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ x ) ) ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ x ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).

thf(1776,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_GT) ).

thf(14092,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1776]) ).

thf(2621,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LTu_conjunct9) ).

thf(18757,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2621]) ).

thf(1013,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_TRIVu_NUMSEG) ).

thf(9171,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1013]) ).

thf(2553,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
    <=> ( p @ ( s @ bool @ f0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_INu_CLAUSESu_conjunct0) ).

thf(18357,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
       => ( p @ ( s @ bool @ f0 ) ) )
      & ( ( p @ ( s @ bool @ f0 ) )
       => ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2553]) ).

thf(2692,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_INV) ).

thf(19090,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2692]) ).

thf(329,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_ADD1) ).

thf(5209,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).

thf(448,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIFFu_INTERS) ).

thf(5930,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(2283,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
    <=> ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_BETWEEN) ).

thf(16886,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
       => ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
      & ( ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2283]) ).

thf(1360,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ f ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ f ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ f ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_CLAUSESu_NUMSEGu_conjunct1) ).

thf(11328,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ f ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ f ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ f ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1360]) ).

thf(1758,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) )
        = ( s @ ( list @ A ) @ nil ) )
    <=> ( ( ( s @ ( list @ A ) @ B )
          = ( s @ ( list @ A ) @ nil ) )
        & ( ( s @ ( list @ A ) @ C )
          = ( s @ ( list @ A ) @ nil ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aAPPENDu_EQu_NIL) ).

thf(13879,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) )
          = ( s @ ( list @ A ) @ nil ) )
       => ( ( ( s @ ( list @ A ) @ B )
            = ( s @ ( list @ A ) @ nil ) )
          & ( ( s @ ( list @ A ) @ C )
            = ( s @ ( list @ A ) @ nil ) ) ) )
      & ( ( ( ( s @ ( list @ A ) @ B )
            = ( s @ ( list @ A ) @ nil ) )
          & ( ( s @ ( list @ A ) @ C )
            = ( s @ ( list @ A ) @ nil ) ) )
       => ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) )
          = ( s @ ( list @ A ) @ nil ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1758]) ).

thf(2311,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ r_r_ ) @ ( s @ real @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ r_r_ ) @ ( s @ real @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_MULu_LMUL) ).

thf(16990,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ r_r_ ) @ ( s @ real @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ r_r_ ) @ ( s @ real @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ C ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2311]) ).

thf(1831,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EQu_conjunct6) ).

thf(14362,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1831]) ).

thf(1609,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ b0 ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ a0 ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ b0 ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_SUBu_RDISTRIB) ).

thf(12913,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ b0 ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ a0 ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ b0 ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1609]) ).

thf(2858,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ E ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXu_IMP) ).

thf(20004,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ E ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2858]) ).

thf(436,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_OFFSETu_0) ).

thf(5887,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).

thf(1692,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_SQUAREu_ABS) ).

thf(13480,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1692]) ).

thf(3003,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_COMM) ).

thf(20793,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3003]) ).

thf(3072,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SGN) ).

thf(21214,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3072]) ).

thf(2560,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_INSEG) ).

thf(18389,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2560]) ).

thf(3114,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
         => ! [E: $i] :
              ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) )
              = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',atransp) ).

thf(21434,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
         => ! [E: $i] :
              ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) )
              = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ D ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3114]) ).

thf(1987,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_ASSOC) ).

thf(15344,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1987]) ).

thf(1386,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( list @ A ) @ B )
       != ( s @ ( list @ A ) @ nil ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ B ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLENGTHu_TL) ).

thf(11539,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( list @ A ) @ B )
       != ( s @ ( list @ A ) @ nil ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ B ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1386]) ).

thf(2523,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ G ) ) ) ) )
         => ( ( s @ A @ F )
            = ( s @ A @ G ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ D ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) ) @ o ) @ ( s @ ( fun @ B @ real ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_IMAGE) ).

thf(18169,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ G ) ) ) ) )
         => ( ( s @ A @ F )
            = ( s @ A @ G ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ D ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ real ) ) ) @ o ) @ ( s @ ( fun @ B @ real ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2523]) ).

thf(2738,axiom,
    ! [A: $i] :
      ( ( ! [B: $i,C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
        & ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSITIVEu_STEPWISEu_LT) ).

thf(19315,plain,
    ! [A: $i] :
      ( ( ! [B: $i,C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
        & ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2738]) ).

thf(2292,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
        & ! [E: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) ) )
           => ( ( s @ A @ E )
              = ( s @ A @ F ) ) )
        & ! [E: $i] :
          ? [F: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) )
            = ( s @ A @ E ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_IMAGE) ).

thf(16923,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
        & ! [E: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) ) )
           => ( ( s @ A @ E )
              = ( s @ A @ F ) ) )
        & ! [E: $i] :
          ? [F: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) )
            = ( s @ A @ E ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2292]) ).

thf(773,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
            & ! [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ E ) @ ( s @ A @ G ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ E ) @ ( s @ A @ H ) ) ) ) )
               => ( ( s @ A @ G )
                  = ( s @ A @ H ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ B ) ) ) @ o ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ A ) @ E ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_INJECTION) ).

thf(7802,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
            & ! [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ E ) @ ( s @ A @ G ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ E ) @ ( s @ A @ H ) ) ) ) )
               => ( ( s @ A @ G )
                  = ( s @ A @ H ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ B ) ) ) @ o ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ A ) @ E ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[773]) ).

thf(1200,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_1u_conjunct3) ).

thf(10190,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1200]) ).

thf(2975,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_TRIANGLE) ).

thf(20614,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2975]) ).

thf(2819,axiom,
    ( ( ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
   => ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWLOGu_LT) ).

thf(19775,plain,
    ( ( ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
   => ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2819]) ).

thf(1028,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_LT) ).

thf(9226,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1028]) ).

thf(2426,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_SUBSET) ).

thf(17646,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2426]) ).

thf(1036,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) )
        <=> ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) ) )
        <=> ? [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_PAIREDu_THM) ).

thf(9256,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) )
           => ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
          & ( ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) ) )
           => ? [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) )
          & ( ? [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1036]) ).

thf(2729,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ ( sum0 @ A @ B ) @ C ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ ( sum0 @ B @ C ) ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ ( sum0 @ A @ B ) @ C ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ ( sum0 @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ A @ ( sum0 @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( sum0 @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ A @ ( sum0 @ B @ C ) ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( sum0 @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ B @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ B @ C ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ ( sum0 @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ ( sum0 @ A @ B ) @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ ( sum0 @ A @ B ) @ C ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_ADDu_ASSOC) ).

thf(19283,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ ( sum0 @ A @ B ) @ C ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ ( sum0 @ B @ C ) ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ ( sum0 @ A @ B ) @ C ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ ( sum0 @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ A @ ( sum0 @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( sum0 @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ A @ ( sum0 @ B @ C ) ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( sum0 @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ B @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ B @ C ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ ( sum0 @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ ( sum0 @ A @ B ) @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ ( sum0 @ A @ B ) @ C ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2729]) ).

thf(1566,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i] :
        ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_NONZERO) ).

thf(12677,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i] :
        ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1566]) ).

thf(2853,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ A @ G ) ) )
          = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ F ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ G ) ) ) @ ( s @ B @ H ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i,I: $i] :
                  ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ F ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ A @ I ) ) )
                  = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ A @ I ) ) ) ) ) )
             => ! [G: $i,H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) ) )
                 => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ F ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ G ) ) ) ) )
                    = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ G ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_SWAP) ).

thf(19977,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ A @ G ) ) )
          = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ F ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ G ) ) ) @ ( s @ B @ H ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i,I: $i] :
                  ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ F ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ A @ I ) ) )
                  = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ A @ I ) ) ) ) ) )
             => ! [G: $i,H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) ) )
                 => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ F ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ G ) ) ) ) )
                    = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ G ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2853]) ).

thf(2662,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
        ? [E: $i] :
          ( ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( list @ A ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( list @ A ) @ E ) ) )
          = ( s @ ( list @ B ) @ D ) )
    <=> ! [D: $i] :
        ? [E: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
          = ( s @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJECTIVEu_MAP) ).

thf(18924,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
          ? [E: $i] :
            ( ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( list @ A ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( list @ A ) @ E ) ) )
            = ( s @ ( list @ B ) @ D ) )
       => ! [D: $i] :
          ? [E: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
            = ( s @ B @ D ) ) )
      & ( ! [D: $i] :
          ? [E: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
            = ( s @ B @ D ) )
       => ! [D: $i] :
          ? [E: $i] :
            ( ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( list @ A ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( list @ A ) @ E ) ) )
            = ( s @ ( list @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2662]) ).

thf(3045,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
     => ( ( ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ B @ F )
                = ( s @ B @ G ) ) )
          & ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
             => ? [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) )
                    = ( s @ A @ F ) ) ) ) )
      <=> ? [F: $i] :
            ( ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ G ) ) ) ) )
                  = ( s @ A @ G ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
               => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
                  = ( s @ B @ G ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIJECTIVEu_ONu_LEFTu_RIGHTu_INVERSE) ).

thf(21023,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
     => ( ( ( ! [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
               => ( ( s @ B @ F )
                  = ( s @ B @ G ) ) )
            & ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ? [G: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) )
                      = ( s @ A @ F ) ) ) ) )
         => ? [F: $i] :
              ( ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
              & ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                 => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ G ) ) ) ) )
                    = ( s @ A @ G ) ) )
              & ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                 => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
                    = ( s @ B @ G ) ) ) ) )
        & ( ? [F: $i] :
              ( ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
              & ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                 => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ G ) ) ) ) )
                    = ( s @ A @ G ) ) )
              & ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                 => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
                    = ( s @ B @ G ) ) ) )
         => ( ! [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
               => ( ( s @ B @ F )
                  = ( s @ B @ G ) ) )
            & ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ? [G: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) )
                      = ( s @ A @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3045]) ).

thf(123,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) )
    <=> ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EQu_SQUAREu_ABS) ).

thf(3807,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(2575,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aGSPEC) ).

thf(18472,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2575]) ).

thf(1265,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ real ) @ E ) ) ) )
        <=> ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ geq ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ real ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) )
            <=> ? [H: $i,I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ J ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ H ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ real ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ H ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SUMu_PRODUCT) ).

thf(10592,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ real ) @ E ) ) ) )
           => ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ geq ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ real ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
          & ( ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ geq ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ real ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ real ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) )
               => ? [H: $i,I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) )
              & ( ? [H: $i,I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ real ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ A @ real ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ H ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ real ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ real ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ real ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ real ) ) @ H ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1265]) ).

thf(286,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aONE) ).

thf(4919,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(422,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i,C: $i] :
        ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LINVu_LEMMA0) ).

thf(5808,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i,C: $i] :
        ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).

thf(2381,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
    <=> ? [C: $i] :
          ( ( s @ hreal @ B )
          = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_LEu_EXISTSu_DEF) ).

thf(17321,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
       => ? [C: $i] :
            ( ( s @ hreal @ B )
            = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) )
      & ( ? [C: $i] :
            ( ( s @ hreal @ B )
            = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2381]) ).

thf(544,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_LTu_SQUAREu_ALT) ).

thf(6469,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).

thf(2177,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ num @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ num @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_RMUL) ).

thf(16343,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ num @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ num @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2177]) ).

thf(287,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ D ) ) ) @ ( s @ hreal @ B ) ) ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ D ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',atrealu_mul) ).

thf(4921,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ D ) ) ) @ ( s @ hreal @ B ) ) ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ D ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(1393,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ max ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAX) ).

thf(11578,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ max ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1393]) ).

thf(1554,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_ADDNEG2) ).

thf(12609,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1554]) ).

thf(2896,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ D ) @ ( cart @ ( cart @ real @ C ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ D ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ D ) @ ( cart @ ( cart @ real @ C ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ D ) @ B ) @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ D ) @ G ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ D ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ D ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ D ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ D ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ D ) @ B ) @ ( cart @ ( cart @ real @ D ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ D ) @ B ) @ ( cart @ ( cart @ real @ D ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ D ) @ B ) @ F ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ D ) @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_MULu_ASSOC) ).

thf(20226,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ D ) @ ( cart @ ( cart @ real @ C ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ D ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ D ) @ ( cart @ ( cart @ real @ C ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ D ) @ B ) @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ D ) @ G ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ D ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ D ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ D ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ D ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ D ) @ B ) @ ( cart @ ( cart @ real @ D ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ D ) @ B ) @ ( cart @ ( cart @ real @ D ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ D ) @ B ) @ F ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ D ) @ G ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2896]) ).

thf(2161,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
        <=> ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFu_EQ) ).

thf(16265,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
           => ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) )
          & ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
           => ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2161]) ).

thf(3097,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_ANTISYM) ).

thf(21349,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3097]) ).

thf(865,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
        <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ G ) ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ F ) ) ) )
            & ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ num @ G ) ) ) ) )
            & ! [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ H )
                   != ( s @ A @ I ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_UNIONS) ).

thf(8333,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
           => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ G ) ) ) ) ) ) )
          & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ F ) ) ) )
            & ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ num @ G ) ) ) ) )
            & ! [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ H )
                   != ( s @ A @ I ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[865]) ).

thf(1807,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( ? [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ D ) @ ( s @ B @ G ) ) )
                    = ( s @ B @ F ) ) )
              & ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ D ) @ ( s @ B @ G ) ) )
                      = ( s @ B @ F ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ D ) @ ( s @ B @ H ) ) )
                      = ( s @ B @ F ) ) )
                 => ( ( s @ B @ G )
                    = ( s @ B @ H ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ B ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ B ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_BIJECTION) ).

thf(14231,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( ? [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ D ) @ ( s @ B @ G ) ) )
                    = ( s @ B @ F ) ) )
              & ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ D ) @ ( s @ B @ G ) ) )
                      = ( s @ B @ F ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ D ) @ ( s @ B @ H ) ) )
                      = ( s @ B @ F ) ) )
                 => ( ( s @ B @ G )
                    = ( s @ B @ H ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ B ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ B ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1807]) ).

thf(3064,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( sum0 @ B @ A ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ B @ A ) ) @ inl ) @ ( s @ B @ C ) ) )
        = ( s @ ( sum0 @ B @ A ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ B @ A ) ) @ inl ) @ ( s @ B @ D ) ) ) )
    <=> ( ( s @ B @ C )
        = ( s @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asumu_INJECTIVEu_conjunct0) ).

thf(21165,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( sum0 @ B @ A ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ B @ A ) ) @ inl ) @ ( s @ B @ C ) ) )
          = ( s @ ( sum0 @ B @ A ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ B @ A ) ) @ inl ) @ ( s @ B @ D ) ) ) )
       => ( ( s @ B @ C )
          = ( s @ B @ D ) ) )
      & ( ( ( s @ B @ C )
          = ( s @ B @ D ) )
       => ( ( s @ ( sum0 @ B @ A ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ B @ A ) ) @ inl ) @ ( s @ B @ C ) ) )
          = ( s @ ( sum0 @ B @ A ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ B @ A ) ) @ inl ) @ ( s @ B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3064]) ).

thf(1940,axiom,
    ( ( ( s @ bool @ a0 )
      = ( s @ bool @ b0 ) )
   => ( ( p @ ( s @ bool @ a0 ) )
     => ( p @ ( s @ bool @ b0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_IMP) ).

thf(15062,plain,
    ( ( ( s @ bool @ a0 )
      = ( s @ bool @ b0 ) )
   => ( ( p @ ( s @ bool @ a0 ) )
     => ( p @ ( s @ bool @ b0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1940]) ).

thf(1404,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        = ( s @ real @ A ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_POW2) ).

thf(11692,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ real @ A ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1404]) ).

thf(2200,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ C ) ) ) )
    <=> ( ( s @ num @ B )
        = ( s @ num @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECu_EQ) ).

thf(16477,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ B ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ C ) ) ) )
       => ( ( s @ num @ B )
          = ( s @ num @ C ) ) )
      & ( ( ( s @ num @ B )
          = ( s @ num @ C ) )
       => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ B ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2200]) ).

thf(2324,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_IMPu_LE) ).

thf(17050,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2324]) ).

thf(380,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_RZERO) ).

thf(5524,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).

thf(552,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSETu_OFu_LISTu_APPEND) ).

thf(6507,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).

thf(2533,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ C @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ! [J: $i,K: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ in ) @ ( s @ C @ H ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ C @ H ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) )
            <=> ? [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
         => ! [F: $i,G: $i] :
              ( ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( fun @ C @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) ) ) ) )
              = ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) @ gspec ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERSu_GSPECu_conjunct1) ).

thf(18255,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ C @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ! [J: $i,K: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ in ) @ ( s @ C @ H ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) ) ) )
                & ( ! [J: $i,K: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ in ) @ ( s @ C @ H ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ C @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ! [J: $i,K: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ in ) @ ( s @ C @ H ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) ) ) )
                & ( ! [J: $i,K: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ in ) @ ( s @ C @ H ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ C @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ C @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) )
               => ? [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
              & ( ? [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ C @ bool ) @ H ) ) ) ) ) )
         => ! [F: $i,G: $i] :
              ( ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( fun @ C @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) ) ) ) )
              = ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) @ gspec ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ C @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2533]) ).

thf(1076,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_LINV) ).

thf(9416,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1076]) ).

thf(1601,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ int @ A )
        = ( s @ int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_0) ).

thf(12874,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ int @ A )
          = ( s @ int @ B ) ) )
      & ( ( ( s @ int @ A )
          = ( s @ int @ B ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1601]) ).

thf(1197,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_EVENu_POW2) ).

thf(10175,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1197]) ).

thf(1118,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_OFu_NUMu_ADD) ).

thf(9765,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1118]) ).

thf(212,axiom,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ l0 ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREVERSEu_conjunct1) ).

thf(4380,plain,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ l0 ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(2248,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
         => ? [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ F ) ) )
                = ( s @ B @ E ) ) ) )
    <=> ? [E: $i] :
        ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
            & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) )
              = ( s @ B @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJECTIVEu_ONu_RIGHTu_INVERSE) ).

thf(16709,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ? [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ F ) ) )
                  = ( s @ B @ E ) ) ) )
       => ? [E: $i] :
          ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) )
                = ( s @ B @ F ) ) ) ) )
      & ( ? [E: $i] :
          ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) )
                = ( s @ B @ F ) ) ) )
       => ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ? [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ F ) ) )
                  = ( s @ B @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2248]) ).

thf(1938,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_REFL) ).

thf(15057,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1938]) ).

thf(2771,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_NEG) ).

thf(19530,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2771]) ).

thf(2277,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ real @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_LMUL) ).

thf(16857,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ real @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2277]) ).

thf(2828,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_ASSOC) ).

thf(19846,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ A ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ C ) @ B ) @ ( cart @ ( cart @ real @ C ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ C ) @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2828]) ).

thf(2029,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) )
    <=> ( ( s @ A @ B )
        = ( s @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_SING) ).

thf(15570,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) )
       => ( ( s @ A @ B )
          = ( s @ A @ C ) ) )
      & ( ( ( s @ A @ B )
          = ( s @ A @ C ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2029]) ).

thf(1085,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ min ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMIN) ).

thf(9502,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ min ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1085]) ).

thf(2063,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMu_COUNTABLE) ).

thf(15707,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2063]) ).

thf(2671,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_LEu_ADDu_LCANCEL) ).

thf(18968,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2671]) ).

thf(2947,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
    <=> ( p @ ( s @ bool @ t ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_INu_CLAUSESu_conjunct0) ).

thf(20490,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
       => ( p @ ( s @ bool @ t ) ) )
      & ( ( p @ ( s @ bool @ t ) )
       => ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2947]) ).

thf(663,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LT2u_REV) ).

thf(7155,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[663]) ).

thf(777,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
     => ? [C: $i] :
          ( ( ( p @ ( s @ bool @ C ) )
          <=> ( ( s @ ( fun @ real @ bool ) @ B )
              = ( s @ ( fun @ real @ bool ) @ empty ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPu_INSERTu_FINITE) ).

thf(7831,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
     => ? [C: $i] :
          ( ( ( p @ ( s @ bool @ C ) )
           => ( ( s @ ( fun @ real @ bool ) @ B )
              = ( s @ ( fun @ real @ bool ) @ empty ) ) )
          & ( ( ( s @ ( fun @ real @ bool ) @ B )
              = ( s @ ( fun @ real @ bool ) @ empty ) )
           => ( p @ ( s @ bool @ C ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[777]) ).

thf(2159,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_SUBSET) ).

thf(16260,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2159]) ).

thf(2206,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_MIN) ).

thf(16505,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2206]) ).

thf(1874,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_LVSUM) ).

thf(14620,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1874]) ).

thf(1487,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_ADD) ).

thf(12119,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1487]) ).

thf(2941,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( ( s @ A @ C )
            = ( s @ A @ D ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNWINDu_THM1) ).

thf(20466,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] :
            ( ( ( s @ A @ C )
              = ( s @ A @ D ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ? [D: $i] :
            ( ( ( s @ A @ C )
              = ( s @ A @ D ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2941]) ).

thf(1650,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
      <=> ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INFu_LEu_FINITE) ).

thf(13259,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
         => ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
        & ( ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1650]) ).

thf(1922,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_NUMSEG) ).

thf(14982,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1922]) ).

thf(2614,axiom,
    ! [A: $i] :
      ( ( s @ n10 @ ( i @ ( s @ ( fun @ bool @ n10 ) @ oneu_abs ) @ ( s @ bool @ ( i @ ( s @ ( fun @ n10 @ bool ) @ oneu_rep ) @ ( s @ n10 @ A ) ) ) ) )
      = ( s @ n10 @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aoneu_tydefu_conjunct0) ).

thf(18709,plain,
    ! [A: $i] :
      ( ( s @ n10 @ ( i @ ( s @ ( fun @ bool @ n10 ) @ oneu_abs ) @ ( s @ bool @ ( i @ ( s @ ( fun @ n10 @ bool ) @ oneu_rep ) @ ( s @ n10 @ A ) ) ) ) )
      = ( s @ n10 @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2614]) ).

thf(893,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
          = ( s @ real @ a0 ) )
      <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ a0 ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
          & ! [B: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFu_UNIQUEu_FINITE) ).

thf(8480,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
            = ( s @ real @ a0 ) )
         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ a0 ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
            & ! [B: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ B ) ) ) ) ) ) )
        & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ a0 ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
            & ! [B: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ B ) ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
            = ( s @ real @ a0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[893]) ).

thf(1568,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ B )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_EQu_1u_IMP) ).

thf(12683,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ B )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1568]) ).

thf(2253,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ f ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ ( fun @ B @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_CLAUSESu_conjunct0) ).

thf(16747,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ f ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ ( fun @ B @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2253]) ).

thf(20,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( finite_image @ A ) @ num ) @ destu_finiteu_image ) @ ( s @ ( finite_image @ A ) @ B ) ) ) ) )
      = ( s @ ( finite_image @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',afiniteu_imageu_tybiju_conjunct0) ).

thf(3262,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( finite_image @ A ) @ num ) @ destu_finiteu_image ) @ ( s @ ( finite_image @ A ) @ B ) ) ) ) )
      = ( s @ ( finite_image @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(1321,axiom,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( p @ ( s @ bool @ D ) )
      <=> ( ( s @ ( list @ A ) @ C )
          = ( s @ ( list @ A ) @ nil ) ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLASTu_APPEND) ).

thf(10993,plain,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( p @ ( s @ bool @ D ) )
       => ( ( s @ ( list @ A ) @ C )
          = ( s @ ( list @ A ) @ nil ) ) )
      & ( ( ( s @ ( list @ A ) @ C )
          = ( s @ ( list @ A ) @ nil ) )
       => ( p @ ( s @ bool @ D ) ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1321]) ).

thf(2758,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ A ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_ANTISYM) ).

thf(19418,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ A ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2758]) ).

thf(463,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_UNIONu_RZERO) ).

thf(6001,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).

thf(1574,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ? [C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_CHOOSEu_SIZE) ).

thf(12704,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ? [C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ real @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1574]) ).

thf(1929,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_RADD) ).

thf(15025,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1929]) ).

thf(2407,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_SUBu_ABS) ).

thf(17563,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2407]) ).

thf(305,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_BOUNDu_COMPONENTu_LE) ).

thf(5016,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(1214,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
    <=> ? [B: $i] :
          ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_EXISTS) ).

thf(10274,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
       => ? [B: $i] :
            ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
      & ( ? [B: $i] :
            ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1214]) ).

thf(2988,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_CONG) ).

thf(20701,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2988]) ).

thf(1665,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ x ) ) )
    = ( s @ int @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MINu_ACIu_conjunct3) ).

thf(13336,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ x ) ) )
    = ( s @ int @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1665]) ).

thf(2489,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ n10 @ A ) @ C ) @ ( s @ n10 @ one ) ) )
          = ( s @ A @ B ) )
      & ! [C: $i,D: $i] :
          ( ( ( ( s @ A @ ( i @ ( s @ ( fun @ n10 @ A ) @ C ) @ ( s @ n10 @ one ) ) )
              = ( s @ A @ B ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ n10 @ A ) @ D ) @ ( s @ n10 @ one ) ) )
              = ( s @ A @ B ) ) )
         => ( ( s @ ( fun @ n10 @ A ) @ C )
            = ( s @ ( fun @ n10 @ A ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aoneu_Axiom) ).

thf(17991,plain,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ n10 @ A ) @ C ) @ ( s @ n10 @ one ) ) )
          = ( s @ A @ B ) )
      & ! [C: $i,D: $i] :
          ( ( ( ( s @ A @ ( i @ ( s @ ( fun @ n10 @ A ) @ C ) @ ( s @ n10 @ one ) ) )
              = ( s @ A @ B ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ n10 @ A ) @ D ) @ ( s @ n10 @ one ) ) )
              = ( s @ A @ B ) ) )
         => ( ( s @ ( fun @ n10 @ A ) @ C )
            = ( s @ ( fun @ n10 @ A ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2489]) ).

thf(460,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
        | ( ( s @ num @ C )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_MULTu_RCANCEL) ).

thf(5979,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ B ) )
          | ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ B ) )
          | ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).

thf(1752,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_UNION) ).

thf(13849,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1752]) ).

thf(2480,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_LMUL) ).

thf(17949,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2480]) ).

thf(2927,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_ASSOC) ).

thf(20384,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2927]) ).

thf(1111,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ! [H: $i] :
                    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ H ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) @ ( s @ num @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ num @ H ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ num @ ( fun @ B @ bool ) ) ) @ D ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ B @ G ) ) ) )
            <=> ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) @ ( s @ num @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ B @ H ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ num @ ( fun @ B @ bool ) ) ) @ D ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_CART) ).

thf(9650,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ! [H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) @ ( s @ num @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ num @ H ) ) ) ) ) ) ) )
                & ( ! [H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) @ ( s @ num @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ num @ H ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ! [H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) @ ( s @ num @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ num @ H ) ) ) ) ) ) ) )
                & ( ! [H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) @ ( s @ num @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ num @ B ) ) @ d_ ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) @ ( s @ num @ H ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ bool @ ( fun @ ( cart @ B @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( cart @ B @ A ) @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( cart @ B @ A ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ num @ ( fun @ B @ bool ) ) ) @ D ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ B @ G ) ) ) )
               => ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) @ ( s @ num @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ B @ H ) ) ) ) )
              & ( ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) @ ( s @ num @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ B @ H ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ num @ ( fun @ B @ bool ) ) ) @ D ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ num @ ( fun @ B @ bool ) ) ) @ D ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( cart @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ B @ bool ) ) @ ( fun @ ( cart @ B @ A ) @ bool ) ) @ C ) @ ( s @ ( fun @ num @ ( fun @ B @ bool ) ) @ E ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1111]) ).

thf(98,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SGNu_NEG) ).

thf(3696,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(549,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_negu_th) ).

thf(6492,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).

thf(2120,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
        <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_EX) ).

thf(16012,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
          & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
          & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
           => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2120]) ).

thf(2531,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) )
        <=> ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
        <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_ALL) ).

thf(18242,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) )
           => ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) ) )
          & ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
          & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
           => ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2531]) ).

thf(2862,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ I ) ) ) ) )
                = ( s @ B @ I ) ) ) )
        & ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) )
                = ( s @ A @ I ) )
              & ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_EQu_GENERALu_INVERSES) ).

thf(20024,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ I ) ) ) ) )
                = ( s @ B @ I ) ) ) )
        & ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ H ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) )
                = ( s @ A @ I ) )
              & ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2862]) ).

thf(2433,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) @ ( s @ B @ E ) ) ) )
        <=> ? [F: $i] :
              ( ( s @ ( prod @ A @ B ) @ D )
              = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ B ) @ snd ) @ ( s @ ( prod @ A @ B ) @ D ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ B ) @ h_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSNDu_DEF) ).

thf(17703,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) @ ( s @ B @ E ) ) ) )
           => ? [F: $i] :
                ( ( s @ ( prod @ A @ B ) @ D )
                = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) ) )
          & ( ? [F: $i] :
                ( ( s @ ( prod @ A @ B ) @ D )
                = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ B ) @ snd ) @ ( s @ ( prod @ A @ B ) @ D ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ B ) @ h_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2433]) ).

thf(708,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        <=> ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_POWERSET) ).

thf(7372,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
          & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[708]) ).

thf(312,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBASISu_EQu_0) ).

thf(5043,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) ) )
      & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(1661,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_MUL) ).

thf(13296,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1661]) ).

thf(2440,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_ADDu_SYM) ).

thf(17748,plain,
    ! [A: $i,B: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2440]) ).

thf(2061,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ bool ) @ B ) @ ( s @ ( recspace @ A ) @ bottom ) ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ bool ) @ B ) @ ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ bool ) @ B ) @ ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ E ) ) ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ bool ) @ B ) @ ( s @ ( recspace @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONSTRu_IND) ).

thf(15696,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ bool ) @ B ) @ ( s @ ( recspace @ A ) @ bottom ) ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ bool ) @ B ) @ ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ bool ) @ B ) @ ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ E ) ) ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( recspace @ A ) @ bool ) @ B ) @ ( s @ ( recspace @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2061]) ).

thf(1706,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_1u_LT) ).

thf(13559,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1706]) ).

thf(944,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_RCANCELu_IMP) ).

thf(8807,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[944]) ).

thf(2979,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_ADD2) ).

thf(20657,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2979]) ).

thf(896,axiom,
    ! [A: $i] :
      ( ( ? [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ? [B: $i] :
          ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
          & ! [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ D ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_COMPLETEu_SOMEPOS) ).

thf(8495,plain,
    ! [A: $i] :
      ( ( ? [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ? [B: $i] :
          ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
          & ! [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ D ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[896]) ).

thf(1238,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_DELETEu_CASES) ).

thf(10425,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1238]) ).

thf(2505,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_INV) ).

thf(18064,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2505]) ).

thf(611,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_REALu_INTERVALu_conjunct4) ).

thf(6870,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).

thf(553,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_LZERO) ).

thf(6510,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).

thf(62,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_HALFu_conjunct0) ).

thf(3519,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(2145,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJOINTu_UNION) ).

thf(16154,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2145]) ).

thf(1945,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ num @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_LMUL) ).

thf(15079,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ num @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1945]) ).

thf(2825,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ u_guardedu_pattern ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) ) @ ( s @ bool @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ A ) )
        & ( p @ ( s @ bool @ B ) )
        & ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',au_GUARDEDu_PATTERN) ).

thf(19814,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ u_guardedu_pattern ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) ) @ ( s @ bool @ C ) ) ) )
       => ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ B ) )
          & ( p @ ( s @ bool @ C ) ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ B ) )
          & ( p @ ( s @ bool @ C ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ u_guardedu_pattern ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) ) @ ( s @ bool @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2825]) ).

thf(2609,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injf ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ B ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numfst ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numsnd ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJF) ).

thf(18689,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injf ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ B ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numfst ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numsnd ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2609]) ).

thf(2254,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ A ) )
    <=> ( ( p @ ( s @ bool @ A ) )
       => ( p @ ( s @ bool @ f0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_DEF) ).

thf(16750,plain,
    ! [A: $i] :
      ( ( ~ ( p @ ( s @ bool @ A ) )
       => ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ f0 ) ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ f0 ) ) )
       => ~ ( p @ ( s @ bool @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2254]) ).

thf(2296,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) )
         => ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_INDUCT) ).

thf(16935,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) ) )
          & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) )
         => ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2296]) ).

thf(1194,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ? [D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
          = ( s @ real @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_CHOOSEu_DIST) ).

thf(10155,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ? [D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
          = ( s @ real @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1194]) ).

thf(251,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
      <=> ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_INFu_FINITE) ).

thf(4662,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
         => ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
        & ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(1666,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_0) ).

thf(13338,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1666]) ).

thf(83,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ D ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_EQu_0) ).

thf(3636,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ D ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(1778,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EVENu_conjunct0) ).

thf(14097,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1778]) ).

thf(763,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ real ) @ neutral ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNEUTRALu_REALu_MUL) ).

thf(7736,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ real ) @ neutral ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[763]) ).

thf(1107,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
        <=> ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) )
            <=> ? [I: $i,J: $i,K: $i] :
                  ( ( ( p @ ( s @ bool @ K ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i,I: $i,J: $i] :
                  ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) ) ) @ F ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ I ) ) ) @ ( s @ A @ J ) ) )
                  = ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ J ) ) ) ) ) @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ I ) @ ( s @ A @ J ) ) ) ) ) )
             => ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) )
                 => ! [H: $i,I: $i,J: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                        & ! [K: $i] :
                            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ K ) ) ) ) ) ) ) )
                     => ( ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) ) ) @ F ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ J ) ) ) ) )
                        = ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ C ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_ITERATEu_PRODUCT) ).

thf(9602,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
           => ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
          & ( ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) )
               => ? [I: $i,J: $i,K: $i] :
                    ( ( ( p @ ( s @ bool @ K ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ K ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) )
              & ( ? [I: $i,J: $i,K: $i] :
                    ( ( ( p @ ( s @ bool @ K ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ K ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i,I: $i,J: $i] :
                  ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) ) ) @ F ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ I ) ) ) @ ( s @ A @ J ) ) )
                  = ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ J ) ) ) ) ) @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ I ) @ ( s @ A @ J ) ) ) ) ) )
             => ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) )
                 => ! [H: $i,I: $i,J: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                        & ! [K: $i] :
                            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ K ) ) ) ) ) ) ) )
                     => ( ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ A @ C ) ) ) ) @ F ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ J ) ) ) ) )
                        = ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ C ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ J ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1107]) ).

thf(2370,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
        = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) )
    <=> ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
        = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSPu_EQ) ).

thf(17280,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
          = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) )
       => ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
          = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
      & ( ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
          = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) )
       => ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
          = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2370]) ).

thf(2539,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_LNEG) ).

thf(18302,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2539]) ).

thf(1517,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
        <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ E ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
            <=> ? [H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ I ) )
                  <=> ! [J: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ H ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ E ) @ ( s @ B @ J ) ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ H ) ) ) ) ) ) ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                <=> ? [I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ G ) @ ( s @ B @ I ) ) ) ) ) ) ) ) )
             => ! [F: $i,G: $i] :
                  ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ F ) ) ) ) ) ) )
                  = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERSu_OVERu_UNIONS) ).

thf(12317,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
           => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ E ) @ ( s @ B @ G ) ) ) ) ) ) )
          & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ E ) @ ( s @ B @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
               => ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ! [J: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ H ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ E ) @ ( s @ B @ J ) ) ) ) ) ) ) )
                    & ( ! [J: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ H ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ E ) @ ( s @ B @ J ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ H ) ) ) ) ) ) ) ) ) ) ) )
              & ( ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ! [J: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ H ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ E ) @ ( s @ B @ J ) ) ) ) ) ) ) )
                    & ( ! [J: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ H ) @ ( s @ B @ J ) ) ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ E ) @ ( s @ B @ J ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ H ) ) ) ) ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                   => ? [I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ G ) @ ( s @ B @ I ) ) ) ) ) ) ) ) )
                  & ( ? [I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ G ) @ ( s @ B @ I ) ) ) ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) ) )
             => ! [F: $i,G: $i] :
                  ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ F ) ) ) ) ) ) )
                  = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1517]) ).

thf(725,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSUM) ).

thf(7487,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[725]) ).

thf(323,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_MULu_RID) ).

thf(5121,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).

thf(1425,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_ADDL) ).

thf(11809,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1425]) ).

thf(1450,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_SUBu_EQ) ).

thf(11936,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ C ) )
       => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1450]) ).

thf(446,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_ADDu_SPLIT) ).

thf(5923,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).

thf(2677,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_EQu_REFL) ).

thf(18990,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2677]) ).

thf(678,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_ABSu_BOUND) ).

thf(7223,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[678]) ).

thf(990,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_0) ).

thf(9025,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[990]) ).

thf(767,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) )
         => ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCHOOSEu_SUBSET) ).

thf(7751,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) )
         => ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[767]) ).

thf(718,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_EQu_0u_DOT) ).

thf(7440,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[718]) ).

thf(2341,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ inter ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ inter ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPPORTu_CLAUSESu_conjunct4) ).

thf(17141,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ inter ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ inter ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2341]) ).

thf(3014,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ bool @ bool ) @ univ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_BOOL) ).

thf(20847,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ bool @ bool ) @ univ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3014]) ).

thf(1681,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SGNu_MUL) ).

thf(13395,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1681]) ).

thf(109,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_NUMSEGu_LT) ).

thf(3758,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(152,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_UNIONu_LZERO) ).

thf(3945,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(2171,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
    <=> ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_EQu_SUBSET) ).

thf(16323,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
       => ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
      & ( ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2171]) ).

thf(254,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EXPu_conjunct2) ).

thf(4677,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(2410,axiom,
    ( ( ( p @ ( s @ bool @ p1 ) )
      | ( p @ ( s @ bool @ p1 ) ) )
  <=> ( p @ ( s @ bool @ p1 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJu_ACIu_conjunct3) ).

thf(17573,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2410]) ).

thf(1547,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
        <=> ! [F: $i,G: $i] :
            ? [H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                  | ( ( ( s @ A @ G )
                      = ( s @ A @ D ) )
                    & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ F ) ) ) )
                      | ( ( s @ A @ F )
                        = ( s @ A @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) @ ( s @ A @ x ) ) ) )
        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ x ) ) ) )
            | ( ( s @ A @ x )
              = ( s @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFLu_SUC) ).

thf(12541,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
           => ! [F: $i,G: $i] :
              ? [H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    | ( ( ( s @ A @ G )
                        = ( s @ A @ D ) )
                      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ F ) ) ) )
                        | ( ( s @ A @ F )
                          = ( s @ A @ D ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    | ( ( ( s @ A @ G )
                        = ( s @ A @ D ) )
                      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ F ) ) ) )
                        | ( ( s @ A @ F )
                          = ( s @ A @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
          & ( ! [F: $i,G: $i] :
              ? [H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    | ( ( ( s @ A @ G )
                        = ( s @ A @ D ) )
                      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ F ) ) ) )
                        | ( ( s @ A @ F )
                          = ( s @ A @ D ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    | ( ( ( s @ A @ G )
                        = ( s @ A @ D ) )
                      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ F ) ) ) )
                        | ( ( s @ A @ F )
                          = ( s @ A @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) @ ( s @ A @ x ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ x ) ) ) )
              | ( ( s @ A @ x )
                = ( s @ A @ D ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ x ) ) ) )
              | ( ( s @ A @ x )
                = ( s @ A @ D ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) @ ( s @ A @ x ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1547]) ).

thf(2566,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_TRIANGLE) ).

thf(18418,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2566]) ).

thf(274,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) )
      = ( s @ ( list @ ( prod @ A @ B ) ) @ nil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZIPu_DEFu_conjunct0) ).

thf(4801,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) )
      = ( s @ ( list @ ( prod @ A @ B ) ) @ nil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(2933,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ C ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_SUBu_RADD) ).

thf(20411,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ C ) )
       => ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2933]) ).

thf(2380,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_DIFF) ).

thf(17315,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2380]) ).

thf(2150,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_ADDu_FINITEu_EQ) ).

thf(16209,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2150]) ).

thf(262,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) ) )
        & ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_BOUNDu_LT) ).

thf(4709,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) ) )
        & ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(1594,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ind0 @ num ) @ mku_num ) @ ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ ( i @ ( s @ ( fun @ num @ ind0 ) @ destu_num ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUCu_DEF) ).

thf(12840,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ind0 @ num ) @ mku_num ) @ ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ ( i @ ( s @ ( fun @ num @ ind0 ) @ destu_num ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1594]) ).

thf(645,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POLYu_NEGu_CLAUSESu_conjunct1) ).

thf(7041,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[645]) ).

thf(2751,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
    <=> ? [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_INu_UNIONS) ).

thf(19381,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
       => ? [D: $i,E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) ) )
      & ( ? [D: $i,E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2751]) ).

thf(243,axiom,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
      <=> ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_ZERO) ).

thf(4605,plain,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
       => ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ B ) ) )
      & ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(313,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_LZERO) ).

thf(5050,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).

thf(2689,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCANTORu_THMu_UNIV) ).

thf(19083,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2689]) ).

thf(1707,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ B ) ) )
        = ( s @ ( fun @ A @ bool ) @ empty ) )
    <=> ( ( s @ ( list @ A ) @ B )
        = ( s @ ( list @ A ) @ nil ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSETu_OFu_LISTu_EQu_EMPTY) ).

thf(13561,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ B ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
       => ( ( s @ ( list @ A ) @ B )
          = ( s @ ( list @ A ) @ nil ) ) )
      & ( ( ( s @ ( list @ A ) @ B )
          = ( s @ ( list @ A ) @ nil ) )
       => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ B ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1707]) ).

thf(754,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ f ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ x ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ x ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ f ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_CLAUSESu_conjunct1) ).

thf(7704,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ f ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ x ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ x ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ f ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[754]) ).

thf(1503,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SGNu_0) ).

thf(12259,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1503]) ).

thf(2734,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ n10 @ bool ) @ A ) @ ( s @ n10 @ one ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ n10 @ bool ) @ A ) @ ( s @ n10 @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aoneu_INDUCT) ).

thf(19296,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ n10 @ bool ) @ A ) @ ( s @ n10 @ one ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ n10 @ bool ) @ A ) @ ( s @ n10 @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2734]) ).

thf(1000,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_01) ).

thf(9057,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1000]) ).

thf(1257,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_SUB) ).

thf(10557,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1257]) ).

thf(76,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
        | ? [D: $i,E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_ZEROu_EXISTS) ).

thf(3604,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
        | ? [D: $i,E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(2234,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ A @ G ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_OP) ).

thf(16649,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ A @ G ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2234]) ).

thf(2942,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_EXISTS) ).

thf(20475,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2942]) ).

thf(2900,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) )
     => ? [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_SUBSETu_IMAGEu_IMP) ).

thf(20237,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) )
     => ? [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2900]) ).

thf(1881,axiom,
    ! [A: $i] :
    ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_ARCH) ).

thf(14642,plain,
    ! [A: $i] :
    ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1881]) ).

thf(1055,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_MULTu_conjunct7) ).

thf(9339,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1055]) ).

thf(244,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
       != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_POSu_LT) ).

thf(4612,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
         != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
         != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(2179,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORDINALu_CHAINED) ).

thf(16349,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2179]) ).

thf(2968,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_NORM) ).

thf(20586,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2968]) ).

thf(2724,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_SUBu_LADD) ).

thf(19264,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2724]) ).

thf(3127,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ A @ E ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ numleft ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numright ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numright ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ A @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJP) ).

thf(21497,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ A @ E ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ numleft ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numright ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numright ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ A @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3127]) ).

thf(2528,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( ( s @ A @ D )
             != ( s @ A @ E ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apairwise) ).

thf(18213,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ! [D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
              & ( ( s @ A @ D )
               != ( s @ A @ E ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
      & ( ! [D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
              & ( ( s @ A @ D )
               != ( s @ A @ E ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2528]) ).

thf(1128,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_EXPu_MOD) ).

thf(9820,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1128]) ).

thf(1467,axiom,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
      <=> ( ( s @ ( list @ A ) @ t0 )
          = ( s @ ( list @ A ) @ nil ) ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ A @ h0 ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLAST) ).

thf(12011,plain,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
       => ( ( s @ ( list @ A ) @ t0 )
          = ( s @ ( list @ A ) @ nil ) ) )
      & ( ( ( s @ ( list @ A ) @ t0 )
          = ( s @ ( list @ A ) @ nil ) )
       => ( p @ ( s @ bool @ B ) ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ A @ h0 ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1467]) ).

thf(1252,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ C ) ) ) ) @ map2 ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ l0 ) ) )
      = ( s @ ( list @ C ) @ nil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAP2u_DEFu_conjunct0) ).

thf(10525,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ C ) ) ) ) @ map2 ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ l0 ) ) )
      = ( s @ ( list @ C ) @ nil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1252]) ).

thf(420,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTWO) ).

thf(5799,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).

thf(1616,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EXPu_conjunct5) ).

thf(12964,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1616]) ).

thf(742,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_GE0) ).

thf(7622,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[742]) ).

thf(2317,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ num @ real ) @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( fun @ num @ real ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_REALu_SEQUENCES) ).

thf(17009,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ num @ real ) @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( fun @ num @ real ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2317]) ).

thf(11,axiom,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) )
      = ( s @ A @ h0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLASTu_CLAUSESu_conjunct0) ).

thf(3213,plain,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) )
      = ( s @ A @ h0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(2639,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_MULu_ASSOC) ).

thf(18812,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2639]) ).

thf(2165,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arow) ).

thf(16291,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2165]) ).

thf(2336,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_REFL) ).

thf(17115,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2336]) ).

thf(1174,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ B ) @ num ) @ length ) @ ( s @ ( list @ B ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ D ) ) ) )
     => ( ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ B @ A ) ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ A ) @ B ) @ ( fun @ ( list @ ( prod @ B @ A ) ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ ( prod @ B @ A ) @ B ) @ fst ) ) ) @ ( s @ ( list @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ ( prod @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ A ) @ ( list @ ( prod @ B @ A ) ) ) ) @ zip0 ) @ ( s @ ( list @ B ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
        = ( s @ ( list @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_FSTu_ZIP) ).

thf(10044,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ B ) @ num ) @ length ) @ ( s @ ( list @ B ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ D ) ) ) )
     => ( ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ B @ A ) ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ A ) @ B ) @ ( fun @ ( list @ ( prod @ B @ A ) ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ ( prod @ B @ A ) @ B ) @ fst ) ) ) @ ( s @ ( list @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ ( prod @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ A ) @ ( list @ ( prod @ B @ A ) ) ) ) @ zip0 ) @ ( s @ ( list @ B ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
        = ( s @ ( list @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1174]) ).

thf(2590,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ADDu_ASSOC) ).

thf(18590,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2590]) ).

thf(2835,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) )
    <=> ( ( s @ int @ A )
        = ( s @ int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EQu_ADDu_RCANCEL) ).

thf(19881,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) )
       => ( ( s @ int @ A )
          = ( s @ int @ B ) ) )
      & ( ( ( s @ int @ A )
          = ( s @ int @ B ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2835]) ).

thf(1672,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_RZERO) ).

thf(13358,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1672]) ).

thf(2468,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) )
    <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aODDu_conjunct1) ).

thf(17905,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) )
      & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2468]) ).

thf(932,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
    <=> ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_EQu_1) ).

thf(8767,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      & ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[932]) ).

thf(1427,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ADDu_conjunct6) ).

thf(11819,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1427]) ).

thf(2726,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_MULT) ).

thf(19269,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2726]) ).

thf(1040,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_POSu_LE) ).

thf(9285,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1040]) ).

thf(1810,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINSEGu_WOSET) ).

thf(14258,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1810]) ).

thf(1775,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
    <=> ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ainseg) ).

thf(14076,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
       => ! [D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) )
            & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ) )
      & ( ! [D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) )
            & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1775]) ).

thf(2322,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_RMUL) ).

thf(17041,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2322]) ).

thf(308,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_1u_LE) ).

thf(5033,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(182,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
        <=> ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ real @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ real @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ real @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ A ) ) ) ) ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_COUNTABLEu_SUBSETSu_REAL) ).

thf(4127,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
           => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ real @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ real @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ real @ bool ) @ C ) ) ) ) )
          & ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ real @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ real @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ real @ bool ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ A ) ) ) ) ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(1795,axiom,
    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ r0 ) ) ) )
  <=> ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ r0 ) ) ) ) )
      = ( s @ real @ r0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_rep) ).

thf(14144,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ r0 ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ r0 ) ) ) ) )
        = ( s @ real @ r0 ) ) )
    & ( ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ r0 ) ) ) ) )
        = ( s @ real @ r0 ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ r0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1795]) ).

thf(108,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ! [G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
               => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_SUPERSET) ).

thf(3751,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ! [G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
               => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(536,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ? [G: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONS) ).

thf(6385,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ? [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                & ( ? [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ? [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                & ( ? [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).

thf(1769,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INFu_BOUNDS) ).

thf(13953,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1769]) ).

thf(2267,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ( ( p @ ( s @ bool @ C ) )
            & ( ( s @ A @ D )
              = ( s @ A @ E ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ E ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ bool @ E ) ) ) @ ( s @ A @ D ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i] :
                  ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ A @ bool ) ) @ D ) @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) )
                  = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) ) ) )
             => ! [E: $i,F: $i] :
                  ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ A @ bool ) ) @ D ) @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) ) ) ) ) ) )
                  = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_ELIMu_THMu_conjunct0) ).

thf(16796,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ( ( p @ ( s @ bool @ C ) )
              & ( ( s @ A @ D )
                = ( s @ A @ E ) ) ) )
          & ( ( ( p @ ( s @ bool @ C ) )
              & ( ( s @ A @ D )
                = ( s @ A @ E ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ E ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ bool @ E ) ) ) @ ( s @ A @ D ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i] :
                  ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ A @ bool ) ) @ D ) @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) )
                  = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) ) ) )
             => ! [E: $i,F: $i] :
                  ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ A @ bool ) ) @ D ) @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) ) ) ) ) ) )
                  = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ ( fun @ A @ bool ) ) @ bool ) @ E ) @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ A @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2267]) ).

thf(2543,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ! [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ E ) ) ) )
                 => ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ F ) ) )
                    = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) )
             => ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) )
                = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
         => ! [C: $i,D: $i] :
              ( ( ! [E: $i] :
                    ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) )
                    = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) ) )
                & ! [E: $i] :
                    ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) )
                    = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
             => ( ( s @ ( fun @ A @ bool ) @ C )
                = ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_URECu_WF) ).

thf(18314,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ! [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ E ) ) ) )
                 => ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ F ) ) )
                    = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) )
             => ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) )
                = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
         => ! [C: $i,D: $i] :
              ( ( ! [E: $i] :
                    ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) )
                    = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ E ) ) ) )
                & ! [E: $i] :
                    ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) )
                    = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
             => ( ( s @ ( fun @ A @ bool ) @ C )
                = ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2543]) ).

thf(2079,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_ADD) ).

thf(15765,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2079]) ).

thf(2174,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ C ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_WF) ).

thf(16334,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ C ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2174]) ).

thf(1648,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
        <=> ! [F: $i,G: $i] :
            ? [H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                  | ( ( ( s @ A @ G )
                      = ( s @ A @ D ) )
                    & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ F ) ) ) )
                      | ( ( s @ A @ F )
                        = ( s @ A @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXTENDu_LINSEG) ).

thf(13217,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
           => ! [F: $i,G: $i] :
              ? [H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    | ( ( ( s @ A @ G )
                        = ( s @ A @ D ) )
                      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ F ) ) ) )
                        | ( ( s @ A @ F )
                          = ( s @ A @ D ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    | ( ( ( s @ A @ G )
                        = ( s @ A @ D ) )
                      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ F ) ) ) )
                        | ( ( s @ A @ F )
                          = ( s @ A @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
          & ( ! [F: $i,G: $i] :
              ? [H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    | ( ( ( s @ A @ G )
                        = ( s @ A @ D ) )
                      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ F ) ) ) )
                        | ( ( s @ A @ F )
                          = ( s @ A @ D ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                    | ( ( ( s @ A @ G )
                        = ( s @ A @ D ) )
                      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ F ) ) ) )
                        | ( ( s @ A @ F )
                          = ( s @ A @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1648]) ).

thf(621,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ o ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALLu_MAP) ).

thf(6937,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ o ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).

thf(530,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_SUBR2) ).

thf(6357,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).

thf(2387,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ~ ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_DISJOINT) ).

thf(17368,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ~ ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      & ( ~ ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2387]) ).

thf(303,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_CLAUSESu_RIGHT) ).

thf(5011,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(2209,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aODDu_MULT) ).

thf(16515,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2209]) ).

thf(2050,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ? [F: $i] :
                ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ F ) ) )
                = ( s @ B @ E ) ) )
        & ! [E: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ f ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ B @ bool ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJECTIVEu_IMAGEu_EQ) ).

thf(15649,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ? [F: $i] :
                ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ F ) ) )
                = ( s @ B @ E ) ) )
        & ! [E: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ f ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ B @ bool ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2050]) ).

thf(856,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
        ? [H: $i] :
          ( ( ( p @ ( s @ bool @ H ) )
          <=> ( ( s @ A @ G )
              = ( s @ A @ D ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ A @ G ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_DELTA) ).

thf(8286,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
        ? [H: $i] :
          ( ( ( p @ ( s @ bool @ H ) )
           => ( ( s @ A @ G )
              = ( s @ A @ D ) ) )
          & ( ( ( s @ A @ G )
              = ( s @ A @ D ) )
           => ( p @ ( s @ bool @ H ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) @ ( s @ A @ G ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ F ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ A @ B ) ) ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[856]) ).

thf(1184,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERSu_2) ).

thf(10083,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1184]) ).

thf(2345,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_COMPONENT) ).

thf(17162,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2345]) ).

thf(2722,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
    <=> ( ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
           => ( ( s @ A @ C )
              = ( s @ A @ D ) ) )
        & ! [C: $i] :
            ( ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
              & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                & ! [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSET) ).

thf(19234,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
       => ( ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) )
          & ! [C: $i] :
              ( ( ! [D: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
                & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
             => ? [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                  & ! [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ) ) ) )
      & ( ( ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) )
          & ! [C: $i] :
              ( ( ! [D: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
                & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
             => ? [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                  & ! [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2722]) ).

thf(962,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_INCLu_EXCL) ).

thf(8862,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[962]) ).

thf(758,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_OFFSETu_0) ).

thf(7719,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[758]) ).

thf(1454,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
        ? [G: $i] :
          ( ( ( p @ ( s @ bool @ G ) )
          <=> ( ( s @ num @ D )
              = ( s @ num @ F ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ G ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ B ) ) ) @ D ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ num @ F ) ) ) @ ( s @ num @ E ) ) ) ) ) )
         => ! [E: $i] :
              ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ A ) ) @ mat ) @ ( s @ num @ E ) ) )
              = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ B ) ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amat) ).

thf(11951,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
        ? [G: $i] :
          ( ( ( p @ ( s @ bool @ G ) )
           => ( ( s @ num @ D )
              = ( s @ num @ F ) ) )
          & ( ( ( s @ num @ D )
              = ( s @ num @ F ) )
           => ( p @ ( s @ bool @ G ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ G ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ B ) ) ) @ D ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ num @ F ) ) ) @ ( s @ num @ E ) ) ) ) ) )
         => ! [E: $i] :
              ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ A ) ) @ mat ) @ ( s @ num @ E ) ) )
              = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ B ) ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1454]) ).

thf(1383,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( ( s @ int @ A )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POSu_NZ) ).

thf(11502,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( ( s @ int @ A )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1383]) ).

thf(2328,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) )
    <=> ( ( s @ hreal @ B )
        = ( s @ hreal @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_EQu_ADDu_LCANCEL) ).

thf(17077,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
          = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) )
       => ( ( s @ hreal @ B )
          = ( s @ hreal @ C ) ) )
      & ( ( ( s @ hreal @ B )
          = ( s @ hreal @ C ) )
       => ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
          = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2328]) ).

thf(921,axiom,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ m ) ) ) @ ( s @ hreal @ n ) ) ) ) ) @ ( s @ hreal @ p1 ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ m ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ n ) ) ) @ ( s @ hreal @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_ADDu_ACu_conjunct1) ).

thf(8665,plain,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ m ) ) ) @ ( s @ hreal @ n ) ) ) ) ) @ ( s @ hreal @ p1 ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ m ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ n ) ) ) @ ( s @ hreal @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[921]) ).

thf(88,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GEu_conjunct7) ).

thf(3663,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(252,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_add) ).

thf(4672,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(2448,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
        <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ F ) ) ) ) )
        <=> ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_INu_GSPECu_conjunct0) ).

thf(17789,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
           => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) ) )
          & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ F ) ) ) ) )
           => ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) ) ) )
          & ( ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) ) )
           => ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2448]) ).

thf(2014,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_SUBSET) ).

thf(15517,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2014]) ).

thf(636,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_LSQRT) ).

thf(6991,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[636]) ).

thf(2297,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ union ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_UNION) ).

thf(16945,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ union ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2297]) ).

thf(1544,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
        ? [H: $i] :
          ( ( ( p @ ( s @ bool @ H ) )
          <=> ( ( s @ B @ G )
              = ( s @ B @ D ) ) )
          & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ F ) ) ) @ ( s @ B @ G ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ F ) ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) ) @ C ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
          = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPPORTu_DELTA) ).

thf(12524,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
        ? [H: $i] :
          ( ( ( p @ ( s @ bool @ H ) )
           => ( ( s @ B @ G )
              = ( s @ B @ D ) ) )
          & ( ( ( s @ B @ G )
              = ( s @ B @ D ) )
           => ( p @ ( s @ bool @ H ) ) )
          & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ F ) ) ) @ ( s @ B @ G ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ F ) ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) ) @ C ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
          = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1544]) ).

thf(2966,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_NEG) ).

thf(20581,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2966]) ).

thf(1223,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_LMUL) ).

thf(10348,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1223]) ).

thf(218,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_LDIVu_EQ) ).

thf(4413,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(1176,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_NORM) ).

thf(10051,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1176]) ).

thf(1400,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LEu_conjunct9) ).

thf(11662,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1400]) ).

thf(1791,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
         => ( ( s @ A @ C )
            = ( s @ A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_ANTISYM) ).

thf(14134,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
         => ( ( s @ A @ C )
            = ( s @ A @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1791]) ).

thf(2641,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_EQu_IMPu_LE) ).

thf(18822,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2641]) ).

thf(895,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_DIVu_REFL) ).

thf(8492,plain,
    ! [A: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[895]) ).

thf(859,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_LT) ).

thf(8300,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[859]) ).

thf(1458,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_RMULu_EQ) ).

thf(11968,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1458]) ).

thf(2270,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ? [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFu_FINITEu_LEMMA) ).

thf(16823,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ? [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2270]) ).

thf(1583,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EXPu_conjunct0) ).

thf(12748,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1583]) ).

thf(2766,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i] :
          ( ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) )
          = ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ u_seqpattern ) @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ H ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) ) ) ) @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i] :
              ( ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ C @ D ) @ K ) ) )
              = ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i,M: $i,N: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                    & ~ ? [O: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) @ ( s @ B @ O ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i,M: $i,N: $i] :
                      ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) )
                      = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) )
                 => ! [K: $i] :
                      ( ! [L: $i,M: $i,N: $i,O: $i] :
                          ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) @ ( s @ E @ O ) ) )
                          = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) @ ( s @ E @ O ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) @ ( s @ E @ O ) ) ) ) ) )
                     => ! [L: $i] :
                          ( ! [M: $i,N: $i,O: $i] :
                              ( ( s @ ( fun @ E @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ N ) ) ) @ ( s @ ( fun @ C @ D ) @ O ) ) )
                              = ( s @ ( fun @ E @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ N ) ) ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) )
                         => ! [M: $i] :
                              ( ! [N: $i,O: $i,P: $i,Q: $i,R: $i] :
                                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ P ) ) ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) )
                                <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) )
                                    & ? [S: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ P ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) ) ) @ ( s @ B @ S ) ) ) ) ) )
                             => ! [N: $i] :
                                  ( ! [O: $i,P: $i,Q: $i,R: $i] :
                                      ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) ) ) @ ( s @ ( fun @ C @ D ) @ R ) ) )
                                      = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) ) ) @ ( s @ ( fun @ C @ D ) @ R ) ) ) )
                                 => ! [O: $i] :
                                      ( ! [P: $i,Q: $i,R: $i,S: $i] :
                                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) ) ) @ ( s @ ( fun @ C @ D ) @ R ) ) ) @ ( s @ E @ S ) ) ) )
                                        <=> ? [T: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ C @ D ) @ R ) ) ) @ ( s @ E @ S ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) @ ( s @ ( fun @ C @ D ) @ R ) ) ) @ ( s @ E @ S ) ) ) ) ) @ ( s @ B @ T ) ) ) ) )
                                     => ! [P: $i] :
                                          ( ! [Q: $i,R: $i,S: $i] :
                                              ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ R ) ) ) @ ( s @ ( fun @ C @ D ) @ S ) ) )
                                              = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ R ) ) ) @ ( s @ ( fun @ C @ D ) @ S ) ) ) )
                                         => ! [Q: $i,R: $i,S: $i,T: $i,U: $i,V: $i] :
                                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ E @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ U ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) ) )
                                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ E @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ U ) ) ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_SEQPATTERN) ).

thf(19466,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i] :
          ( ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) )
          = ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ u_seqpattern ) @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ H ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) ) ) ) @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i] :
              ( ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ C @ D ) @ K ) ) )
              = ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i,M: $i,N: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                   => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                      & ~ ? [O: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) @ ( s @ B @ O ) ) ) ) ) )
                  & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                      & ~ ? [O: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) @ ( s @ B @ O ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i,M: $i,N: $i] :
                      ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) )
                      = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) )
                 => ! [K: $i] :
                      ( ! [L: $i,M: $i,N: $i,O: $i] :
                          ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) @ ( s @ E @ O ) ) )
                          = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) @ ( s @ E @ O ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) @ ( s @ E @ O ) ) ) ) ) )
                     => ! [L: $i] :
                          ( ! [M: $i,N: $i,O: $i] :
                              ( ( s @ ( fun @ E @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ N ) ) ) @ ( s @ ( fun @ C @ D ) @ O ) ) )
                              = ( s @ ( fun @ E @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ N ) ) ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) )
                         => ! [M: $i] :
                              ( ! [N: $i,O: $i,P: $i,Q: $i,R: $i] :
                                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ P ) ) ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) )
                                   => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) )
                                      & ? [S: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ P ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) ) ) @ ( s @ B @ S ) ) ) ) ) )
                                  & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) )
                                      & ? [S: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ P ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) ) ) @ ( s @ B @ S ) ) ) ) )
                                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ P ) ) ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) @ ( s @ E @ R ) ) ) ) ) )
                             => ! [N: $i] :
                                  ( ! [O: $i,P: $i,Q: $i,R: $i] :
                                      ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) ) ) @ ( s @ ( fun @ C @ D ) @ R ) ) )
                                      = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) ) ) @ ( s @ ( fun @ C @ D ) @ R ) ) ) )
                                 => ! [O: $i] :
                                      ( ! [P: $i,Q: $i,R: $i,S: $i] :
                                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) ) ) @ ( s @ ( fun @ C @ D ) @ R ) ) ) @ ( s @ E @ S ) ) ) )
                                           => ? [T: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ C @ D ) @ R ) ) ) @ ( s @ E @ S ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) @ ( s @ ( fun @ C @ D ) @ R ) ) ) @ ( s @ E @ S ) ) ) ) ) @ ( s @ B @ T ) ) ) ) )
                                          & ( ? [T: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ C @ D ) @ R ) ) ) @ ( s @ E @ S ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) @ ( s @ ( fun @ C @ D ) @ R ) ) ) @ ( s @ E @ S ) ) ) ) ) @ ( s @ B @ T ) ) ) )
                                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) ) ) @ ( s @ ( fun @ C @ D ) @ R ) ) ) @ ( s @ E @ S ) ) ) ) ) )
                                     => ! [P: $i] :
                                          ( ! [Q: $i,R: $i,S: $i] :
                                              ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ R ) ) ) @ ( s @ ( fun @ C @ D ) @ S ) ) )
                                              = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ O ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ R ) ) ) @ ( s @ ( fun @ C @ D ) @ S ) ) ) )
                                         => ! [Q: $i,R: $i,S: $i,T: $i,U: $i,V: $i] :
                                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ E @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ P ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ U ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) ) )
                                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ R ) ) ) @ ( s @ ( fun @ E @ A ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ ( fun @ B @ bool ) ) ) ) @ U ) ) ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2766]) ).

thf(2973,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GEu_conjunct0) ).

thf(20609,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2973]) ).

thf(2572,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i,I: $i,J: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ ( fun @ E @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ F ) @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ G ) @ ( s @ D @ J ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) )
     => ! [G: $i] :
          ( ! [H: $i,I: $i,J: $i] :
              ( ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ ( fun @ E @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ G ) @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
              = ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ ( fun @ E @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ F ) @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) )
         => ! [H: $i,I: $i,J: $i,K: $i,L: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ ( fun @ E @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ G ) @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_RAND) ).

thf(18444,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i,I: $i,J: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ ( fun @ E @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ F ) @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) ) ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ G ) @ ( s @ D @ J ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) )
     => ! [G: $i] :
          ( ! [H: $i,I: $i,J: $i] :
              ( ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ ( fun @ E @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ G ) @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) )
              = ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ ( fun @ E @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ F ) @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) )
         => ! [H: $i,I: $i,J: $i,K: $i,L: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ ( fun @ E @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ A ) ) ) ) @ G ) @ ( s @ ( fun @ D @ ( fun @ E @ A ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2572]) ).

thf(2472,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_TOTAL) ).

thf(17918,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2472]) ).

thf(2009,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ y ) ) ) @ ( s @ real @ x ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_ACIu_conjunct0) ).

thf(15485,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ y ) ) ) @ ( s @ real @ x ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2009]) ).

thf(722,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
        <=> ? [G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ? [I: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ I ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
            <=> ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_GSPECu_conjunct0) ).

thf(7455,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
           => ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ I ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ I ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
          & ( ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ I ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ I ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
               => ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) )
              & ( ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) )
              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[722]) ).

thf(2096,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOIDALu_REALu_MUL) ).

thf(15887,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2096]) ).

thf(79,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_SUBu_NORM) ).

thf(3619,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(383,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_DELETE) ).

thf(5539,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(1460,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ A ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_NUMSEGu_0) ).

thf(11973,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ A ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1460]) ).

thf(84,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ num @ B )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_LEu_EXCLUSION) ).

thf(3641,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ num @ B )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(2039,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) )
    <=> ( ( s @ int @ A )
        = ( s @ int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_ANTISYM) ).

thf(15609,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) )
       => ( ( s @ int @ A )
          = ( s @ int @ B ) ) )
      & ( ( ( s @ int @ A )
          = ( s @ int @ B ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2039]) ).

thf(1985,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ C ) ) )
    <=> ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        & ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_EXISTSu_ANDu_THM) ).

thf(15329,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ C ) ) )
       => ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ C ) ) ) )
      & ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ C ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1985]) ).

thf(2491,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_SYM) ).

thf(18003,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2491]) ).

thf(779,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_CLAUSESu_RIGHT) ).

thf(7843,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[779]) ).

thf(1042,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ D ) ) )
      = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ inja ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injf ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZCONSTR) ).

thf(9290,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ D ) ) )
      = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ inja ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injf ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1042]) ).

thf(510,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
        <=> ? [G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_PRODUCT) ).

thf(6248,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) )
           => ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) )
          & ( ? [G: $i,H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( prod @ A @ B ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).

thf(3018,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_OFu_NUMu_EQ) ).

thf(20864,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3018]) ).

thf(566,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_POSu_LE) ).

thf(6549,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).

thf(524,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_INVu_0) ).

thf(6334,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).

thf(2263,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( recspace @ B ) @ A ) @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ num @ A ) ) ) @ C ) @ ( s @ ( fun @ ( recspace @ B ) @ A ) @ D ) ) ) @ ( s @ ( fun @ num @ ( recspace @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( recspace @ B ) @ A ) @ D ) @ ( s @ ( recspace @ B ) @ ( i @ ( s @ ( fun @ num @ ( recspace @ B ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) )
     => ! [D: $i] :
        ? [E: $i] :
        ! [F: $i,G: $i,H: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( recspace @ B ) @ A ) @ E ) @ ( s @ ( recspace @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( recspace @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( recspace @ B ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( recspace @ B ) ) ) ) @ constr ) @ ( s @ num @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ num @ ( recspace @ B ) ) @ H ) ) ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) ) @ D ) @ ( s @ num @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ num @ ( recspace @ B ) ) @ H ) ) ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( recspace @ B ) @ A ) @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ num @ A ) ) ) @ C ) @ ( s @ ( fun @ ( recspace @ B ) @ A ) @ E ) ) ) @ ( s @ ( fun @ num @ ( recspace @ B ) ) @ H ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONSTRu_REC) ).

thf(16782,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( recspace @ B ) @ A ) @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ num @ A ) ) ) @ C ) @ ( s @ ( fun @ ( recspace @ B ) @ A ) @ D ) ) ) @ ( s @ ( fun @ num @ ( recspace @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( recspace @ B ) @ A ) @ D ) @ ( s @ ( recspace @ B ) @ ( i @ ( s @ ( fun @ num @ ( recspace @ B ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) )
     => ! [D: $i] :
        ? [E: $i] :
        ! [F: $i,G: $i,H: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( recspace @ B ) @ A ) @ E ) @ ( s @ ( recspace @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( recspace @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( recspace @ B ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( recspace @ B ) ) ) ) @ constr ) @ ( s @ num @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ num @ ( recspace @ B ) ) @ H ) ) ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ B @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) ) @ D ) @ ( s @ num @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ num @ ( recspace @ B ) ) @ H ) ) ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( recspace @ B ) @ A ) @ ( fun @ ( fun @ num @ ( recspace @ B ) ) @ ( fun @ num @ A ) ) ) @ C ) @ ( s @ ( fun @ ( recspace @ B ) @ A ) @ E ) ) ) @ ( s @ ( fun @ num @ ( recspace @ B ) ) @ H ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2263]) ).

thf(2952,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ A @ C ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ A ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ A @ A ) @ B ) ) ) @ ( s @ ( fun @ A @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aISOu_REFL) ).

thf(20518,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ A @ C ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ A ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ A @ A ) @ B ) ) ) @ ( s @ ( fun @ A @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2952]) ).

thf(2866,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_SYM) ).

thf(20041,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2866]) ).

thf(2447,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_CMULu_SUBu_RDISTRIB) ).

thf(17786,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2447]) ).

thf(1918,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_EQu_TRANS) ).

thf(14942,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1918]) ).

thf(2371,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ? [E: $i] :
        ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_BOUNDED) ).

thf(17287,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ? [E: $i] :
        ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2371]) ).

thf(1416,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_NORMu_SUB) ).

thf(11765,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1416]) ).

thf(30,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h2 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) )
      = ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) ) @ cons ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ h2 ) ) ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZIPu_conjunct1) ).

thf(3302,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h2 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) )
      = ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) ) @ cons ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ h2 ) ) ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(642,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) )
                & ! [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
          & ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZL) ).

thf(7021,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) )
                & ! [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
          & ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[642]) ).

thf(2415,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJOINTu_SYM) ).

thf(17583,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2415]) ).

thf(1288,axiom,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
      <=> ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ f ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_CLAUSESu_NUMSEGu_conjunct0) ).

thf(10741,plain,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
       => ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ B ) ) )
      & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ f ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1288]) ).

thf(2992,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_ADD2u_ABSORBu_LE) ).

thf(20724,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2992]) ).

thf(2064,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ F ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ E ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i] :
              ( ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ G ) ) )
              = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ G ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) ) ) )
                & ! [J: $i,K: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ B @ K ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ H ) @ ( s @ B @ K ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_NEST) ).

thf(15708,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ F ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ E ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i] :
              ( ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ G ) ) )
              = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ G ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) ) ) )
                & ! [J: $i,K: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ B @ K ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ H ) @ ( s @ B @ K ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2064]) ).

thf(2845,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJOINT) ).

thf(19930,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2845]) ).

thf(1966,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_NEGu_WELLDEF) ).

thf(15205,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1966]) ).

thf(2578,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ bij ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ inj ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ surj ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIJ) ).

thf(18487,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ bij ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ inj ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ surj ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ inj ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ surj ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ bij ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2578]) ).

thf(1082,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ invertible ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) )
    <=> ? [D: $i] :
          ( ( ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
            = ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) )
            = ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ainvertible) ).

thf(9483,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ invertible ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) )
       => ? [D: $i] :
            ( ( ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) )
              = ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) )
              = ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ invertible ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1082]) ).

thf(2763,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GTu_conjunct9) ).

thf(19453,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2763]) ).

thf(2102,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
    <=> ( ( s @ int @ B )
        = ( s @ int @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EQu_ADDu_LCANCEL) ).

thf(15941,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
       => ( ( s @ int @ B )
          = ( s @ int @ C ) ) )
      & ( ( ( s @ int @ B )
          = ( s @ int @ C ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2102]) ).

thf(399,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_conjunct1) ).

thf(5634,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).

thf(2776,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALLu_APPEND) ).

thf(19570,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2776]) ).

thf(2205,axiom,
    ( ( ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
   => ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_WLOGu_LT) ).

thf(16500,plain,
    ( ( ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
   => ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ p0 ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2205]) ).

thf(1701,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SGNu_EQu_conjunct0) ).

thf(13531,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1701]) ).

thf(388,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_sgnu_th) ).

thf(5556,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(2805,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ? [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
    <=> ? [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_SUBSETu_IMAGE) ).

thf(19702,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
       => ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) )
      & ( ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) )
       => ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2805]) ).

thf(2813,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ A @ B )
        = ( s @ A @ B ) )
    <=> ( p @ ( s @ bool @ t ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREFLu_CLAUSE) ).

thf(19741,plain,
    p @ ( s @ bool @ t ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2813]) ).

thf(2229,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_0) ).

thf(16632,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2229]) ).

thf(2538,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_UNIONS) ).

thf(18299,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2538]) ).

thf(487,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) )
    <=> ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_SQUAREu_ABS) ).

thf(6144,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).

thf(2892,axiom,
    ! [A: $i,B: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LTu_REFL) ).

thf(20205,plain,
    ! [A: $i,B: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2892]) ).

thf(1621,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ bool @ bool ) @ num ) @ card ) @ ( s @ ( fun @ bool @ bool ) @ univ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_BOOL) ).

thf(13005,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ bool @ bool ) @ num ) @ card ) @ ( s @ ( fun @ bool @ bool ) @ univ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1621]) ).

thf(1863,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_UNIONu_conjunct0) ).

thf(14555,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1863]) ).

thf(2808,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_SQUARE) ).

thf(19715,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2808]) ).

thf(2611,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ bool @ f0 ) )
     => ! [C: $i] :
          ( ! [D: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ ( fun @ A @ bool ) @ B ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_FALSE) ).

thf(18695,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ bool @ f0 ) )
     => ! [C: $i] :
          ( ! [D: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ ( fun @ A @ bool ) @ B ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2611]) ).

thf(546,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_MULT) ).

thf(6478,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).

thf(467,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( cart @ real @ A ) ) @ neutral ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNEUTRALu_VECTORu_ADD) ).

thf(6024,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( cart @ real @ A ) ) @ neutral ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(2864,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJOINTu_DELETEu_SYM) ).

thf(20036,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2864]) ).

thf(1057,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POLYu_CLAUSESu_conjunct9) ).

thf(9344,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1057]) ).

thf(1479,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPREu_conjunct0) ).

thf(12082,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1479]) ).

thf(715,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) )
    <=> ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
        | ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_EXP) ).

thf(7415,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
          | ( ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( ( s @ num @ B )
             != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( ( s @ num @ C )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
          | ( ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( ( s @ num @ B )
             != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( ( s @ num @ C )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[715]) ).

thf(873,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_1u_conjunct4) ).

thf(8408,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[873]) ).

thf(2506,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( ( s @ int @ A )
         != ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_LE) ).

thf(18067,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
          & ( ( s @ int @ A )
           != ( s @ int @ B ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
          & ( ( s @ int @ A )
           != ( s @ int @ B ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2506]) ).

thf(1526,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_DIVu_POW2u_ALT) ).

thf(12428,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1526]) ).

thf(854,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPREu_conjunct1) ).

thf(8276,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[854]) ).

thf(2128,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i,L: $i,M: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) ) ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) @ ( s @ E @ M ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ u_unguardedu_pattern ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ H ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) @ ( s @ E @ M ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) @ ( s @ E @ M ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ F @ bool ) ) @ geq ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) @ ( s @ E @ M ) ) ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) @ ( s @ E @ M ) ) ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i,L: $i,M: $i] :
              ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) )
              = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i,M: $i,N: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) )
                      = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i,M: $i,N: $i] :
                      ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) )
                      = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) )
                 => ! [K: $i,L: $i,M: $i,N: $i,O: $i,P: $i,Q: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ P ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_UNGUARDEDu_PATTERN) ).

thf(16088,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i,L: $i,M: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) ) ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) @ ( s @ E @ M ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ u_unguardedu_pattern ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ H ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) @ ( s @ E @ M ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) @ ( s @ E @ M ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ F @ bool ) ) @ geq ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) @ ( s @ E @ M ) ) ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) @ ( s @ E @ M ) ) ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i,L: $i,M: $i] :
              ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) )
              = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i,M: $i,N: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                   => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) )
                        = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) )
                  & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) )
                        = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i,M: $i,N: $i] :
                      ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) )
                      = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) )
                 => ! [K: $i,L: $i,M: $i,N: $i,O: $i,P: $i,Q: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ P ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ Q ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2128]) ).

thf(3133,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ bool @ t ) )
     => ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ fnil ) @ ( s @ num @ C ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFNIL) ).

thf(21527,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ bool @ t ) )
     => ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ fnil ) @ ( s @ num @ C ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3133]) ).

thf(2282,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',avectoru_norm) ).

thf(16883,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2282]) ).

thf(260,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ m ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ m ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_ofu_num) ).

thf(4704,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ m ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ m ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(290,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ( ( s @ A @ E )
                  = ( s @ A @ C ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSINGu_GSPECu_conjunct0) ).

thf(4929,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( s @ A @ E )
                    = ( s @ A @ C ) ) )
                & ( ( ( s @ A @ E )
                    = ( s @ A @ C ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( s @ A @ E )
                    = ( s @ A @ C ) ) )
                & ( ( ( s @ A @ E )
                    = ( s @ A @ C ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(2062,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_UNION) ).

thf(15700,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2062]) ).

thf(1995,axiom,
    ! [A: $i,B: $i] :
      ( ~ ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
    <=> ? [C: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_FORALLu_THM) ).

thf(15406,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ? [C: $i] :
            ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
      & ( ? [C: $i] :
            ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ~ ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1995]) ).

thf(2648,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ B ) )
     => ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) )
      <=> ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ real @ n10 ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ n10 ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_REALu_ONE) ).

thf(18849,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ B ) )
     => ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) )
         => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ real @ n10 ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ n10 ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) )
        & ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ real @ n10 ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ n10 ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) )
         => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ real @ n10 ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2648]) ).

thf(2398,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_TRIANGLEu_LE) ).

thf(17490,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2398]) ).

thf(2643,axiom,
    ( ( ( p @ ( s @ bool @ p1 ) )
      | ( p @ ( s @ bool @ p1 ) )
      | ( p @ ( s @ bool @ q ) ) )
  <=> ( ( p @ ( s @ bool @ p1 ) )
      | ( p @ ( s @ bool @ q ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJu_ACIu_conjunct4) ).

thf(18827,plain,
    ( ( ( ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ q ) ) )
     => ( ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ q ) ) ) )
    & ( ( ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ q ) ) )
     => ( ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ p1 ) )
        | ( p @ ( s @ bool @ q ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2643]) ).

thf(1953,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_ADDNEG2) ).

thf(15113,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1953]) ).

thf(1239,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_MULu_EQu_conjunct0) ).

thf(10429,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1239]) ).

thf(1418,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) ) ) )
    <=> ( ( s @ num @ ( i @ ( s @ ( fun @ ( finite_sum @ A @ B ) @ num ) @ destu_finiteu_sum ) @ ( s @ ( finite_sum @ A @ B ) @ ( i @ ( s @ ( fun @ num @ ( finite_sum @ A @ B ) ) @ mku_finiteu_sum ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ num @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',afiniteu_sumu_tybiju_conjunct1) ).

thf(11771,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ ( finite_sum @ A @ B ) @ num ) @ destu_finiteu_sum ) @ ( s @ ( finite_sum @ A @ B ) @ ( i @ ( s @ ( fun @ num @ ( finite_sum @ A @ B ) ) @ mku_finiteu_sum ) @ ( s @ num @ C ) ) ) ) )
          = ( s @ num @ C ) ) )
      & ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( finite_sum @ A @ B ) @ num ) @ destu_finiteu_sum ) @ ( s @ ( finite_sum @ A @ B ) @ ( i @ ( s @ ( fun @ num @ ( finite_sum @ A @ B ) ) @ mku_finiteu_sum ) @ ( s @ num @ C ) ) ) ) )
          = ( s @ num @ C ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1418]) ).

thf(572,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) ) ) )
        <=> ! [I: $i,J: $i] :
            ? [K: $i] :
              ( ( ( p @ ( s @ bool @ K ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ A @ I ) ) ) )
                  | ( ( ( s @ A @ E )
                      = ( s @ A @ I ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) @ ( s @ B @ G ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) ) ) )
            <=> ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_LEX) ).

thf(6613,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) ) ) )
           => ! [I: $i,J: $i] :
              ? [K: $i] :
                ( ( ( p @ ( s @ bool @ K ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ A @ I ) ) ) )
                    | ( ( ( s @ A @ E )
                        = ( s @ A @ I ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) @ ( s @ B @ G ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ A @ I ) ) ) )
                    | ( ( ( s @ A @ E )
                        = ( s @ A @ I ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) @ ( s @ B @ G ) ) ) @ ( s @ B @ J ) ) ) ) ) )
                 => ( p @ ( s @ bool @ K ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
          & ( ! [I: $i,J: $i] :
              ? [K: $i] :
                ( ( ( p @ ( s @ bool @ K ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ A @ I ) ) ) )
                    | ( ( ( s @ A @ E )
                        = ( s @ A @ I ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) @ ( s @ B @ G ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ A @ I ) ) ) )
                    | ( ( ( s @ A @ E )
                        = ( s @ A @ I ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) @ ( s @ B @ G ) ) ) @ ( s @ B @ J ) ) ) ) ) )
                 => ( p @ ( s @ bool @ K ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ H ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) ) ) )
               => ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) ) )
              & ( ! [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ F ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).

thf(3165,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_ADD) ).

thf(21664,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3165]) ).

thf(1915,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
    <=> ( ( s @ real @ B )
        = ( s @ real @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_ADDu_LCANCEL) ).

thf(14878,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
       => ( ( s @ real @ B )
          = ( s @ real @ C ) ) )
      & ( ( ( s @ real @ B )
          = ( s @ real @ C ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1915]) ).

thf(493,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_PAIRu_0) ).

thf(6179,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).

thf(3111,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDELETEu_INTER) ).

thf(21424,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ A @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3111]) ).

thf(1495,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
          <=> ( ( s @ real @ B )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_GPu_OFFSET) ).

thf(12227,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
           => ( ( s @ real @ B )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          & ( ( ( s @ real @ B )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ E ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1495]) ).

thf(318,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_DIVu_1) ).

thf(5103,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).

thf(942,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [B: $i] :
        ? [C: $i,D: $i] :
        ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LINVu_LEMMA7a) ).

thf(8799,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [B: $i] :
        ? [C: $i,D: $i] :
        ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[942]) ).

thf(1784,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_P) ).

thf(14119,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1784]) ).

thf(793,axiom,
    ( ? [A: $i] :
        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ p0 ) @ ( s @ int @ A ) ) ) ) )
  <=> ? [A: $i] :
        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ p0 ) @ ( s @ int @ A ) ) ) )
        & ! [B: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ p0 ) @ ( s @ int @ B ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_WOP) ).

thf(7907,plain,
    ( ( ? [A: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ p0 ) @ ( s @ int @ A ) ) ) ) )
     => ? [A: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ p0 ) @ ( s @ int @ A ) ) ) )
          & ! [B: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ p0 ) @ ( s @ int @ B ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) )
    & ( ? [A: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ p0 ) @ ( s @ int @ A ) ) ) )
          & ! [B: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ p0 ) @ ( s @ int @ B ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
     => ? [A: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ p0 ) @ ( s @ int @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).

thf(1181,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) )
          = ( s @ bool @ t ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ E ) @ ( s @ ( fun @ C @ B ) @ F ) ) )
              = ( s @ ( fun @ A @ bool ) @ D ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) ) @ F ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ H ) ) ) )
                <=> ! [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ J ) ) ) ) ) ) )
             => ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) ) @ F ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ G ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ H ) ) ) ) )
                 => ? [I: $i] :
                    ! [J: $i] :
                      ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ I ) @ ( s @ C @ J ) ) )
                      = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ C @ B ) @ I ) ) ) @ ( s @ C @ J ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_RECu_CASESi_) ).

thf(10066,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) )
          = ( s @ bool @ t ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ E ) @ ( s @ ( fun @ C @ B ) @ F ) ) )
              = ( s @ ( fun @ A @ bool ) @ D ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) ) @ F ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ H ) ) ) )
                   => ! [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ J ) ) ) ) ) ) )
                  & ( ! [I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ J ) ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) ) @ F ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ H ) ) ) ) ) )
             => ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) ) @ F ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ G ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ H ) ) ) ) )
                 => ? [I: $i] :
                    ! [J: $i] :
                      ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ I ) @ ( s @ C @ J ) ) )
                      = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ C @ B ) @ I ) ) ) @ ( s @ C @ J ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1181]) ).

thf(827,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_SUBu_DISTRIB) ).

thf(8135,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[827]) ).

thf(1068,axiom,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ m ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ n ) ) ) @ ( s @ hreal @ p1 ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ n ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ m ) ) ) @ ( s @ hreal @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_ADDu_ACu_conjunct2) ).

thf(9377,plain,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ m ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ n ) ) ) @ ( s @ hreal @ p1 ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ n ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ m ) ) ) @ ( s @ hreal @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1068]) ).

thf(155,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIT1u_DEF) ).

thf(3957,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(503,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_CARDu_ADDu_conjunct1) ).

thf(6217,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( sum0 @ A @ B ) @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).

thf(821,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_ADD2) ).

thf(8074,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[821]) ).

thf(2555,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_IDEMPOT) ).

thf(18367,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2555]) ).

thf(1439,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_LT) ).

thf(11878,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1439]) ).

thf(115,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_TRIANGLEu_ADD) ).

thf(3779,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(740,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_DELETE) ).

thf(7607,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[740]) ).

thf(352,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_UNIQ) ).

thf(5303,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).

thf(699,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_POSu_BOUND) ).

thf(7328,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[699]) ).

thf(2502,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
        & ! [F: $i,G: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_IMAGEu_GALOIS) ).

thf(18045,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
        & ! [F: $i,G: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2502]) ).

thf(101,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) )
      = ( s @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSeq2o_HD) ).

thf(3732,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) )
      = ( s @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(1704,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ union ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_COMBINEu_R) ).

thf(13546,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ union ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1704]) ).

thf(2059,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXu_MEM) ).

thf(15686,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2059]) ).

thf(887,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_0) ).

thf(8461,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[887]) ).

thf(2727,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_RNEG) ).

thf(19276,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2727]) ).

thf(1018,axiom,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULTIPLICATIVE) ).

thf(9185,plain,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1018]) ).

thf(883,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LID) ).

thf(8445,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[883]) ).

thf(1741,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        | ( ( s @ num @ C )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_MONOu_LE) ).

thf(13794,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          | ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          | ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1741]) ).

thf(2114,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_SGN) ).

thf(15988,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2114]) ).

thf(700,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EXPu_conjunct3) ).

thf(7330,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[700]) ).

thf(2028,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        | ( p @ ( s @ bool @ C ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_ORu_EXISTSu_THM) ).

thf(15562,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ( p @ ( s @ bool @ C ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            | ( p @ ( s @ bool @ C ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            | ( p @ ( s @ bool @ C ) ) )
       => ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ( p @ ( s @ bool @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2028]) ).

thf(2932,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_DIFFu_FINITE) ).

thf(20407,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2932]) ).

thf(1759,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LETu_ADD) ).

thf(13888,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1759]) ).

thf(3166,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_EQu_SYM) ).

thf(21667,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3166]) ).

thf(1099,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ filter0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ o ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFILTERu_MAP) ).

thf(9570,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ filter0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ o ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1099]) ).

thf(707,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
        <=> ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ D ) ) )
              = ( s @ ( fun @ A @ bool ) @ C ) )
            & ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ D ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( list @ A ) ) @ h_ ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alistu_ofu_set) ).

thf(7360,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
           => ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ D ) ) )
                = ( s @ ( fun @ A @ bool ) @ C ) )
              & ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ D ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
          & ( ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ D ) ) )
                = ( s @ ( fun @ A @ bool ) @ C ) )
              & ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ D ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( list @ A ) ) @ h_ ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[707]) ).

thf(674,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_ACIu_conjunct1) ).

thf(7206,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[674]) ).

thf(1823,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ C @ B ) @ E ) ) ) @ ( s @ C @ F ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ E ) @ ( s @ C @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aou_DEF) ).

thf(14322,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ C @ B ) @ E ) ) ) @ ( s @ C @ F ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ E ) @ ( s @ C @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1823]) ).

thf(987,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_RDIVu_EQ) ).

thf(9015,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[987]) ).

thf(489,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LTu_conjunct5) ).

thf(6154,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).

thf(59,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) ) )
        <=> ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) ) ) )
     => ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ x ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ahrealu_inv) ).

thf(3500,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) ) )
           => ? [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) ) ) )
          & ( ? [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) ) ) ) )
     => ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ x ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(3160,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
        | ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_ORu_EXISTSu_THM) ).

thf(21642,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( p @ ( s @ bool @ B ) )
          | ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ( p @ ( s @ bool @ B ) )
          | ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3160]) ).

thf(1217,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREVERSEu_APPEND) ).

thf(10296,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1217]) ).

thf(2465,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_SUPPORT) ).

thf(17886,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2465]) ).

thf(2119,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ x ) ) )
    = ( s @ real @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_ACIu_conjunct3) ).

thf(16010,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ x ) ) )
    = ( s @ real @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2119]) ).

thf(495,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_INV) ).

thf(6184,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).

thf(2065,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ B @ C )
      = ( s @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aABSu_SIMP) ).

thf(15713,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2065]) ).

thf(1605,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_1) ).

thf(12894,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1605]) ).

thf(2154,axiom,
    ! [A: $i] :
      ( ( s @ ( recspace @ A ) @ bottom )
      = ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( recspace @ A ) ) @ u_mku_rec ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBOTTOM) ).

thf(16231,plain,
    ! [A: $i] :
      ( ( s @ ( recspace @ A ) @ bottom )
      = ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( recspace @ A ) ) @ u_mku_rec ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2154]) ).

thf(499,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTEu_ADD) ).

thf(6196,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).

thf(940,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_DIV2u_EQ) ).

thf(8794,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[940]) ).

thf(649,axiom,
    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) )
  <=> ? [A: $i,B: $i] :
        ( ( ( ( ( s @ num @ n )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( ( s @ num @ B )
              = ( s @ num @ m ) ) )
          | ( ( ( s @ num @ m )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ n ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVMODu_ELIMu_THMi_) ).

thf(7053,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) )
     => ? [A: $i,B: $i] :
          ( ( ( ( ( s @ num @ n )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ B )
                = ( s @ num @ m ) ) )
            | ( ( ( s @ num @ m )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ n ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    & ( ? [A: $i,B: $i] :
          ( ( ( ( ( s @ num @ n )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ B )
                = ( s @ num @ m ) ) )
            | ( ( ( s @ num @ m )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ n ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ m ) ) ) @ ( s @ num @ n ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[649]) ).

thf(2051,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ let ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ B @ D ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLETu_DEF) ).

thf(15654,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ let ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ B @ D ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2051]) ).

thf(225,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) )
        & ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_LT) ).

thf(4477,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) )
        & ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(1372,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ahull) ).

thf(11406,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1372]) ).

thf(2030,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_BOUND) ).

thf(15577,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2030]) ).

thf(1477,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
        <=> ? [G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ? [I: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ B @ G )
                      = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ I ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_IMAGEu_EXPAND) ).

thf(12064,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
           => ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( ( s @ B @ G )
                        = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ I ) ) ) ) ) )
                & ( ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( ( s @ B @ G )
                        = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ I ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
          & ( ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( ( s @ B @ G )
                        = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ I ) ) ) ) ) )
                & ( ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( ( s @ B @ G )
                        = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ I ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1477]) ).

thf(347,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_CROSS) ).

thf(5267,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).

thf(3089,axiom,
    ! [A: $i] :
      ( ( ( ( p @ ( s @ bool @ t ) )
         => ( p @ ( s @ bool @ A ) ) )
      <=> ( p @ ( s @ bool @ A ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ t ) ) )
      <=> ( p @ ( s @ bool @ t ) ) )
      & ( ( ( p @ ( s @ bool @ f0 ) )
         => ( p @ ( s @ bool @ A ) ) )
      <=> ( p @ ( s @ bool @ t ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ A ) ) )
      <=> ( p @ ( s @ bool @ t ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ f0 ) ) )
      <=> ~ ( p @ ( s @ bool @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMPu_CLAUSES) ).

thf(21280,plain,
    ! [A: $i] :
      ( ( ( ( p @ ( s @ bool @ t ) )
         => ( p @ ( s @ bool @ A ) ) )
       => ( p @ ( s @ bool @ A ) ) )
      & ( ( p @ ( s @ bool @ A ) )
       => ( ( p @ ( s @ bool @ t ) )
         => ( p @ ( s @ bool @ A ) ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ t ) ) )
       => ( p @ ( s @ bool @ t ) ) )
      & ( ( p @ ( s @ bool @ t ) )
       => ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ t ) ) ) )
      & ( ( ( p @ ( s @ bool @ f0 ) )
         => ( p @ ( s @ bool @ A ) ) )
       => ( p @ ( s @ bool @ t ) ) )
      & ( ( p @ ( s @ bool @ t ) )
       => ( ( p @ ( s @ bool @ f0 ) )
         => ( p @ ( s @ bool @ A ) ) ) )
      & ( p @ ( s @ bool @ t ) )
      & ( ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ f0 ) ) )
       => ~ ( p @ ( s @ bool @ A ) ) )
      & ( ~ ( p @ ( s @ bool @ A ) )
       => ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ f0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3089]) ).

thf(2922,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPu_HULL) ).

thf(20347,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2922]) ).

thf(694,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_addu_th) ).

thf(7307,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[694]) ).

thf(631,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aODDu_DOUBLE) ).

thf(6974,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[631]) ).

thf(2660,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aANDu_FORALLu_THM) ).

thf(18912,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2660]) ).

thf(2010,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_conjunct1) ).

thf(15487,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2010]) ).

thf(1228,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADD2u_SUB2) ).

thf(10380,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1228]) ).

thf(1571,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_LT2) ).

thf(12691,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1571]) ).

thf(2663,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_DELETE) ).

thf(18931,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2663]) ).

thf(1073,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUCu_conjunct2) ).

thf(9407,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1073]) ).

thf(629,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ n10 @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ n10 @ bool ) @ univ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIMINDEXu_1) ).

thf(6969,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ n10 @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ n10 @ bool ) @ univ ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).

thf(1211,axiom,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ A @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( list @ A ) ) ) @ replicate ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ A @ x ) ) )
      = ( s @ ( list @ A ) @ nil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREPLICATEu_conjunct0) ).

thf(10265,plain,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ A @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( list @ A ) ) ) @ replicate ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ A @ x ) ) )
      = ( s @ ( list @ A ) @ nil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1211]) ).

thf(13,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ numu_gcd ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ int @ num ) @ numu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_gcd) ).

thf(3219,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ numu_gcd ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ int @ num ) @ numu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(386,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLETu_ADD2) ).

thf(5552,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).

thf(2768,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_SUBSETu_conjunct0) ).

thf(19523,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2768]) ).

thf(968,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_POW2) ).

thf(8932,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[968]) ).

thf(2878,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ i1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_I) ).

thf(20147,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ i1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2878]) ).

thf(1499,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ADDu_conjunct7) ).

thf(12248,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1499]) ).

thf(217,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
        <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ A @ G ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ! [F: $i,G: $i] :
                ( ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) )
               => ( ( s @ A @ F )
                  = ( s @ A @ G ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_IMAGEu_INJ) ).

thf(4401,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
           => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ A @ G ) ) ) ) )
          & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ A @ G ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ! [F: $i,G: $i] :
                ( ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) )
               => ( ( s @ A @ F )
                  = ( s @ A @ G ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(911,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
            & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
         => ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
              & ! [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                 => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_WELLu_CONTRAPOS) ).

thf(8620,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
            & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
         => ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
              & ! [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                 => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[911]) ).

thf(223,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_DOWN2) ).

thf(4430,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(1009,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
            <=> ? [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ H ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ E ) @ ( s @ A @ G ) ) )
                       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) ) ) ) ) ) ) )
             => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_ADDu_GEN) ).

thf(9129,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
               => ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ E ) @ ( s @ A @ G ) ) )
                         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ E ) @ ( s @ A @ G ) ) )
                         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
              & ( ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ E ) @ ( s @ A @ G ) ) )
                         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ E ) @ ( s @ A @ G ) ) )
                         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) ) ) ) ) ) ) )
             => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ real ) @ E ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1009]) ).

thf(1436,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SGNu_ABS) ).

thf(11868,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1436]) ).

thf(309,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_CARDu_IMP) ).

thf(5035,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(328,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_SUPERSET) ).

thf(5202,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).

thf(2309,axiom,
    ( ~ ( p @ ( s @ bool @ t ) )
  <=> ( p @ ( s @ bool @ f0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_CLAUSESu_WEAKu_conjunct0) ).

thf(16984,plain,
    ( ( ~ ( p @ ( s @ bool @ t ) )
     => ( p @ ( s @ bool @ f0 ) ) )
    & ( ( p @ ( s @ bool @ f0 ) )
     => ~ ( p @ ( s @ bool @ t ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2309]) ).

thf(2872,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( s @ ( cart @ real @ ( finite_sum @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) ) @ pastecart ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
        = ( s @ ( cart @ real @ ( finite_sum @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) ) @ pastecart ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) )
    <=> ( ( ( s @ ( cart @ real @ A ) @ C )
          = ( s @ ( cart @ real @ A ) @ E ) )
        & ( ( s @ ( cart @ real @ B ) @ D )
          = ( s @ ( cart @ real @ B ) @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPASTECARTu_INJ) ).

thf(20112,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( s @ ( cart @ real @ ( finite_sum @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) ) @ pastecart ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
          = ( s @ ( cart @ real @ ( finite_sum @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) ) @ pastecart ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) )
       => ( ( ( s @ ( cart @ real @ A ) @ C )
            = ( s @ ( cart @ real @ A ) @ E ) )
          & ( ( s @ ( cart @ real @ B ) @ D )
            = ( s @ ( cart @ real @ B ) @ F ) ) ) )
      & ( ( ( ( s @ ( cart @ real @ A ) @ C )
            = ( s @ ( cart @ real @ A ) @ E ) )
          & ( ( s @ ( cart @ real @ B ) @ D )
            = ( s @ ( cart @ real @ B ) @ F ) ) )
       => ( ( s @ ( cart @ real @ ( finite_sum @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) ) @ pastecart ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
          = ( s @ ( cart @ real @ ( finite_sum @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ ( finite_sum @ A @ B ) ) ) ) @ pastecart ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2872]) ).

thf(1785,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ADDu_WELLDEF) ).

thf(14122,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1785]) ).

thf(445,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( list @ A ) @ B )
       != ( s @ ( list @ A ) @ nil ) )
     => ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ butlast ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) )
        = ( s @ ( list @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aAPPENDu_BUTLASTu_LAST) ).

thf(5920,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( list @ A ) @ B )
       != ( s @ ( list @ A ) @ nil ) )
     => ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ butlast ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) )
        = ( s @ ( list @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).

thf(2140,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ j ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ j ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_CMULu_COMPONENT) ).

thf(16139,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ j ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ j ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2140]) ).

thf(2042,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
          = ( s @ ( fun @ A @ bool ) @ B ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) )
         => ( ( s @ A @ D )
            = ( s @ A @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_IMPu_INJECTIVE) ).

thf(15626,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
          = ( s @ ( fun @ A @ bool ) @ B ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) )
         => ( ( s @ A @ D )
            = ( s @ A @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2042]) ).

thf(2212,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ B ) )
        | ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_FORALLu_ORu_THM) ).

thf(16535,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ( p @ ( s @ bool @ B ) )
          | ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ B ) )
          | ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2212]) ).

thf(692,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ H ) ) ) )
        <=> ! [I: $i,J: $i] :
            ? [K: $i] :
              ( ( ( p @ ( s @ bool @ K ) )
              <=> ! [L: $i] :
                    ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ I ) @ ( s @ C @ L ) ) ) ) )
                    = ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ J ) @ ( s @ B @ G ) ) ) @ ( s @ C @ L ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ C @ A ) @ I ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
     => ! [F: $i,G: $i] :
          ( ! [H: $i,I: $i,J: $i,K: $i,L: $i,M: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ C @ A ) @ H ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ I ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ K ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ H ) @ ( s @ C @ L ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ J ) @ ( s @ C @ M ) ) ) ) )
             => ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ I ) @ ( s @ B @ G ) ) ) @ ( s @ C @ L ) ) )
                = ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ K ) @ ( s @ B @ G ) ) ) @ ( s @ C @ M ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCASEWISEu_WORKS) ).

thf(7275,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ H ) ) ) )
           => ! [I: $i,J: $i] :
              ? [K: $i] :
                ( ( ( p @ ( s @ bool @ K ) )
                 => ! [L: $i] :
                      ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ I ) @ ( s @ C @ L ) ) ) ) )
                      = ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ J ) @ ( s @ B @ G ) ) ) @ ( s @ C @ L ) ) ) ) )
                & ( ! [L: $i] :
                      ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ I ) @ ( s @ C @ L ) ) ) ) )
                      = ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ J ) @ ( s @ B @ G ) ) ) @ ( s @ C @ L ) ) ) )
                 => ( p @ ( s @ bool @ K ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ C @ A ) @ I ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) ) )
          & ( ! [I: $i,J: $i] :
              ? [K: $i] :
                ( ( ( p @ ( s @ bool @ K ) )
                 => ! [L: $i] :
                      ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ I ) @ ( s @ C @ L ) ) ) ) )
                      = ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ J ) @ ( s @ B @ G ) ) ) @ ( s @ C @ L ) ) ) ) )
                & ( ! [L: $i] :
                      ( ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ I ) @ ( s @ C @ L ) ) ) ) )
                      = ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ J ) @ ( s @ B @ G ) ) ) @ ( s @ C @ L ) ) ) )
                 => ( p @ ( s @ bool @ K ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ H ) @ ( s @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ C @ A ) @ I ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ J ) ) ) ) ) ) ) @ ( s @ bool @ K ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ H ) ) ) ) ) )
     => ! [F: $i,G: $i] :
          ( ! [H: $i,I: $i,J: $i,K: $i,L: $i,M: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ C @ A ) @ H ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ I ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ ( fun @ C @ D ) ) @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ K ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ H ) @ ( s @ C @ L ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ J ) @ ( s @ C @ M ) ) ) ) )
             => ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ I ) @ ( s @ B @ G ) ) ) @ ( s @ C @ L ) ) )
                = ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ K ) @ ( s @ B @ G ) ) ) @ ( s @ C @ M ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ ( fun @ B @ ( fun @ ( fun @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ bool ) @ bool ) ) ) @ E ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ C @ A ) @ ( fun @ B @ ( fun @ C @ D ) ) ) ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[692]) ).

thf(1764,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ ( fun @ ( cart @ real @ n20 ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ n20 ) @ x ) ) ) @ ( s @ ( cart @ real @ n20 ) @ y ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n20 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n20 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n20 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n20 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_2) ).

thf(13930,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ ( fun @ ( cart @ real @ n20 ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ n20 ) @ x ) ) ) @ ( s @ ( cart @ real @ n20 ) @ y ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n20 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n20 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n20 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n20 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n20 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1764]) ).

thf(2492,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LTu_CONG) ).

thf(18006,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2492]) ).

thf(1227,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ A @ int ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
        & ! [D: $i] :
            ( ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ A @ int ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ A @ int ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_INTu_MEASURE) ).

thf(10375,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ A @ int ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
        & ! [D: $i] :
            ( ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ A @ int ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ A @ int ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1227]) ).

thf(278,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) @ ( s @ ( cart @ real @ A ) @ H ) ) )
          = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ H ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ G ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_VMULu_COMPONENT) ).

thf(4882,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) @ ( s @ ( cart @ real @ A ) @ H ) ) )
          = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ H ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ G ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(1978,axiom,
    ! [A: $i] :
      ( ~ ~ ( p @ ( s @ bool @ A ) )
    <=> ( p @ ( s @ bool @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_CLAUSESu_conjunct0) ).

thf(15280,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1978]) ).

thf(1510,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
        & ? [C: $i] :
          ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUPu_LEu_SUBSET) ).

thf(12284,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
        & ? [C: $i] :
          ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1510]) ).

thf(1485,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LTu_conjunct8) ).

thf(12114,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1485]) ).

thf(1143,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_LNEG) ).

thf(9888,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1143]) ).

thf(1209,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ D ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INFu_ASCLOSE) ).

thf(10253,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ D ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1209]) ).

thf(413,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_DELETE) ).

thf(5730,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).

thf(1423,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ B @ E )
           != ( s @ B @ F ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ A @ G ) ) ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ G ) ) ) ) ) ) )
     => ? [E: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) )
            = ( s @ A @ D ) )
          & ! [F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
             => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSETu_RECURSIONu_LEMMA) ).

thf(11796,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ B @ E )
           != ( s @ B @ F ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ A @ G ) ) ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ G ) ) ) ) ) ) )
     => ? [E: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) )
            = ( s @ A @ D ) )
          & ! [F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
             => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ E ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1423]) ).

thf(1821,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
    <=> ? [D: $i,E: $i] :
          ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
          & ( ( s @ ( list @ A ) @ C )
            = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMu_APPENDu_DECOMPOSEu_LEFT) ).

thf(14311,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
       => ? [D: $i,E: $i] :
            ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
            & ( ( s @ ( list @ A ) @ C )
              = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) ) ) )
      & ( ? [D: $i,E: $i] :
            ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
            & ( ( s @ ( list @ A ) @ C )
              = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1821]) ).

thf(3053,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ E ) ) ) )
        <=> ! [F: $i,G: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ G ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ E ) @ ( s @ ( cart @ real @ B ) @ G ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) )
          = ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ h_ ) @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aadjoint) ).

thf(21099,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ E ) ) ) )
           => ! [F: $i,G: $i] :
                ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ G ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ E ) @ ( s @ ( cart @ real @ B ) @ G ) ) ) ) ) ) )
          & ( ! [F: $i,G: $i] :
                ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ G ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ E ) @ ( s @ ( cart @ real @ B ) @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) )
          = ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ h_ ) @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3053]) ).

thf(2218,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_MAX) ).

thf(16570,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2218]) ).

thf(2175,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ rows ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ columns ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aROWSu_TRANSP) ).

thf(16338,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ rows ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( fun @ ( cart @ real @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ B ) @ bool ) ) @ columns ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2175]) ).

thf(1014,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_ADD) ).

thf(9174,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1014]) ).

thf(2857,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSPu_MATRIXu_SUB) ).

thf(20001,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2857]) ).

thf(3115,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEMPTYu_SUBSET) ).

thf(21437,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3115]) ).

thf(737,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) @ ( s @ ( list @ A ) @ C ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aAPPENDu_SING) ).

thf(7566,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) @ ( s @ ( list @ A ) @ C ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[737]) ).

thf(2348,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_POW) ).

thf(17169,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2348]) ).

thf(3152,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSETu_OFu_LISTu_OFu_SET) ).

thf(21607,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( list @ A ) ) @ listu_ofu_set ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3152]) ).

thf(356,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_SUB) ).

thf(5335,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).

thf(70,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) @ ( s @ num @ F ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_COMPONENT) ).

thf(3572,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) @ ( s @ num @ F ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(295,axiom,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n20 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n20 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ y ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ A @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_2u_conjunct0) ).

thf(4985,plain,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ n20 ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ n20 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n20 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ y ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ A @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(1484,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_CLAUSESu_conjunct2) ).

thf(12111,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1484]) ).

thf(1772,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) @ ( s @ ( cart @ real @ A ) @ n ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ n ) ) ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_ACu_conjunct0) ).

thf(14063,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) @ ( s @ ( cart @ real @ A ) @ n ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ n ) ) ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1772]) ).

thf(1792,axiom,
    ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMiscu_defsu_andu_lemmaso_numu_infinite) ).

thf(14137,plain,
    ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1792]) ).

thf(2704,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ A )
        = ( s @ real @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_IMPu_LE) ).

thf(19146,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ A )
        = ( s @ real @ B ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2704]) ).

thf(1295,axiom,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
      <=> ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_ZERO) ).

thf(10778,plain,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
       => ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ B ) ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1295]) ).

thf(733,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIT1) ).

thf(7551,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[733]) ).

thf(573,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) )
    <=> ( ( ( s @ real @ B )
          = ( s @ real @ C ) )
        | ( ( s @ ( cart @ real @ A ) @ D )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_RCANCEL) ).

thf(6645,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) )
       => ( ( ( s @ real @ B )
            = ( s @ real @ C ) )
          | ( ( s @ ( cart @ real @ A ) @ D )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      & ( ( ( ( s @ real @ B )
            = ( s @ real @ C ) )
          | ( ( s @ ( cart @ real @ A ) @ D )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).

thf(795,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ E ) @ ( s @ D @ F ) ) )
          = ( s @ bool @ t ) )
     => ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ nil ) ) ) @ ( s @ B @ f ) ) ) @ ( s @ C @ x ) ) )
        = ( s @ D @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ D ) @ h_ ) @ ( s @ ( fun @ D @ bool ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCASEWISEu_DEFu_conjunct0) ).

thf(7928,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ E ) @ ( s @ D @ F ) ) )
          = ( s @ bool @ t ) )
     => ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ nil ) ) ) @ ( s @ B @ f ) ) ) @ ( s @ C @ x ) ) )
        = ( s @ D @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ D ) @ h_ ) @ ( s @ ( fun @ D @ bool ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).

thf(804,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ B ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ B ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_CONSTu_NUMSEG) ).

thf(7978,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ B ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ B ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[804]) ).

thf(1582,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_ADDu_SPLIT) ).

thf(12744,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1582]) ).

thf(2710,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_ADD2) ).

thf(19185,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2710]) ).

thf(1529,axiom,
    ( ( s @ num @ u_0 )
    = ( s @ num @ ( i @ ( s @ ( fun @ ind0 @ num ) @ mku_num ) @ ( s @ ind0 @ indu_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZEROu_DEF) ).

thf(12438,plain,
    ( ( s @ num @ u_0 )
    = ( s @ num @ ( i @ ( s @ ( fun @ ind0 @ num ) @ mku_num ) @ ( s @ ind0 @ indu_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1529]) ).

thf(2105,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_RZERO) ).

thf(15961,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2105]) ).

thf(203,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) @ ( s @ num @ H ) ) )
          = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ H ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ G ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) ) ) @ ( s @ num @ I ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ G ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) @ ( s @ num @ I ) ) ) ) ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i,J: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) )
             => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ J ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ J ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ I ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_VSUMu_PARTIALu_SUC) ).

thf(4332,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) @ ( s @ num @ H ) ) )
          = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ H ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ H ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ G ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) ) ) @ ( s @ num @ I ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ G ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ H ) @ ( s @ num @ I ) ) ) ) ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i,J: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) )
             => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ J ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ J ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) @ ( s @ num @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) @ ( s @ num @ I ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ J ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ C ) ) ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ G ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ C ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(4,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ int @ num ) @ numu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMu_OFu_INTu_OFu_NUM) ).

thf(3192,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ int @ num ) @ numu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(3125,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_min) ).

thf(21488,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3125]) ).

thf(588,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ C ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ C ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ C ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ C ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_MULu_LZERO) ).

thf(6747,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ C ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ C ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ C ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ C ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ C ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).

thf(77,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ m ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ m ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_ofu_numu_th) ).

thf(3614,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ m ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(368,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( s @ ( sum0 @ A @ B ) @ C )
          = ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ D ) ) ) )
      | ? [D: $i] :
          ( ( s @ ( sum0 @ A @ B ) @ C )
          = ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asumu_CASES) ).

thf(5466,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( s @ ( sum0 @ A @ B ) @ C )
          = ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ B ) ) @ inl ) @ ( s @ A @ D ) ) ) )
      | ? [D: $i] :
          ( ( s @ ( sum0 @ A @ B ) @ C )
          = ( s @ ( sum0 @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ A @ B ) ) @ inr ) @ ( s @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).

thf(2504,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ inja ) @ ( s @ A @ B ) ) )
        = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ inja ) @ ( s @ A @ C ) ) ) )
    <=> ( ( s @ A @ B )
        = ( s @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJAu_INJ) ).

thf(18057,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ inja ) @ ( s @ A @ B ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ inja ) @ ( s @ A @ C ) ) ) )
       => ( ( s @ A @ B )
          = ( s @ A @ C ) ) )
      & ( ( ( s @ A @ B )
          = ( s @ A @ C ) )
       => ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ inja ) @ ( s @ A @ B ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ inja ) @ ( s @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2504]) ).

thf(2220,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_GT) ).

thf(16585,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2220]) ).

thf(1698,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_LZERO) ).

thf(13518,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1698]) ).

thf(1149,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
      <=> ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_BASIS) ).

thf(9905,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ C ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
        & ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ C ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1149]) ).

thf(473,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ c ) ) ) @ ( s @ A @ x ) ) ) @ ( s @ A @ y ) ) ) ) ) )
    <=> ( ( ( p @ ( s @ bool @ c ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ x ) ) ) ) )
        & ( ~ ( p @ ( s @ bool @ c ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ y ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONDu_ELIMu_THM) ).

thf(6044,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ c ) ) ) @ ( s @ A @ x ) ) ) @ ( s @ A @ y ) ) ) ) ) )
       => ( ( ( p @ ( s @ bool @ c ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ x ) ) ) ) )
          & ( ~ ( p @ ( s @ bool @ c ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ y ) ) ) ) ) ) )
      & ( ( ( ( p @ ( s @ bool @ c ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ x ) ) ) ) )
          & ( ~ ( p @ ( s @ bool @ c ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ y ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ c ) ) ) @ ( s @ A @ x ) ) ) @ ( s @ A @ y ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).

thf(2413,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ A @ B )
      = ( s @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_REFL) ).

thf(17579,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2413]) ).

thf(2131,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_MAX) ).

thf(16101,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2131]) ).

thf(720,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_MUL) ).

thf(7450,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[720]) ).

thf(916,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_NEGL) ).

thf(8652,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[916]) ).

thf(1478,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_RZERO) ).

thf(12079,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1478]) ).

thf(2300,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_SUBu_NUMSEG) ).

thf(16952,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2300]) ).

thf(2107,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_DIV) ).

thf(15966,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2107]) ).

thf(1168,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_NEGR) ).

thf(10004,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1168]) ).

thf(1629,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_conjunct0) ).

thf(13069,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1629]) ).

thf(2098,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_NEGu_ADD) ).

thf(15891,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2098]) ).

thf(3153,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_DELETE) ).

thf(21610,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3153]) ).

thf(1507,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ E ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_EQu_0) ).

thf(12274,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ E ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1507]) ).

thf(875,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) ) ) )
        <=> ! [G: $i,H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ e_ ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) ) ) ) ) )
        <=> ! [F: $i,G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_TRIPLEDu_THM) ).

thf(8413,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) ) ) )
           => ! [G: $i,H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) )
          & ( ! [G: $i,H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ F ) ) ) ) ) )
     => ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ e_ ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) ) ) ) ) )
           => ! [F: $i,G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) )
          & ( ! [F: $i,G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ e_ ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ E ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[875]) ).

thf(1053,axiom,
    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ n ) ) ) ) ) )
  <=> ? [A: $i] :
        ( ( ( ( s @ num @ n )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) )
          | ( ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( ( s @ num @ n )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPREu_ELIMu_THMi_) ).

thf(9327,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ n ) ) ) ) ) )
     => ? [A: $i] :
          ( ( ( ( s @ num @ n )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) )
            | ( ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ n )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) )
    & ( ? [A: $i] :
          ( ( ( ( s @ num @ n )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) )
            | ( ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ n )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ n ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1053]) ).

thf(791,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i] :
              ( ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ num ) ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ ( fun @ num @ num ) @ B ) )
         => ! [D: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ num @ num ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ num ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ num @ num ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARD) ).

thf(7901,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i] :
              ( ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ num ) ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ ( fun @ num @ num ) @ B ) )
         => ! [D: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ num @ num ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ num ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ num @ num ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[791]) ).

thf(2521,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( recspace @ A ) ) @ u_mku_rec ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) )
        = ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( recspace @ A ) ) @ u_mku_rec ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) ) )
       => ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMKu_RECu_INJ) ).

thf(18156,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( recspace @ A ) ) @ u_mku_rec ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) )
        = ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( recspace @ A ) ) @ u_mku_rec ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) ) )
       => ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2521]) ).

thf(551,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_INTER) ).

thf(6504,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).

thf(1022,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
        = ( s @ int @ C ) )
    <=> ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EQu_SUBu_RADD) ).

thf(9197,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ C ) )
       => ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) )
      & ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1022]) ).

thf(876,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_SUBu_LZERO) ).

thf(8423,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[876]) ).

thf(1233,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ C )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_MONO2) ).

thf(10402,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ C )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1233]) ).

thf(2250,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_ACIu_conjunct4) ).

thf(16731,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2250]) ).

thf(1567,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_UNIONu_GEN) ).

thf(12680,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1567]) ).

thf(3131,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
    <=> ( ( s @ ( fun @ B @ bool ) @ C )
        = ( s @ ( fun @ B @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_EMPTY) ).

thf(21513,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
       => ( ( s @ ( fun @ B @ bool ) @ C )
          = ( s @ ( fun @ B @ bool ) @ empty ) ) )
      & ( ( ( s @ ( fun @ B @ bool ) @ C )
          = ( s @ ( fun @ B @ bool ) @ empty ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3131]) ).

thf(2011,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ B ) ) ) ) )
      = ( s @ ( list @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREVERSEu_REVERSE) ).

thf(15496,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ B ) ) ) ) )
      = ( s @ ( list @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2011]) ).

thf(1755,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ADD) ).

thf(13867,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1755]) ).

thf(3143,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAXu_LE) ).

thf(21565,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3143]) ).

thf(1299,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLENGTHu_conjunct0) ).

thf(10793,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1299]) ).

thf(1277,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_MULT2) ).

thf(10660,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1277]) ).

thf(266,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SGN) ).

thf(4720,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(881,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_POSu_BOUND) ).

thf(8440,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[881]) ).

thf(1346,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_PREu_conjunct3) ).

thf(11220,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1346]) ).

thf(2939,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
         => ( ( s @ B @ E )
            = ( s @ B @ F ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_IMAGEu_INJu_EQ) ).

thf(20455,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
         => ( ( s @ B @ E )
            = ( s @ B @ F ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2939]) ).

thf(1623,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
          <=> ( ( s @ A @ D )
              = ( s @ A @ C ) ) )
          & ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ num ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ num ) ) @ B ) @ ( s @ A @ D ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_DELTA) ).

thf(13010,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
           => ( ( s @ A @ D )
              = ( s @ A @ C ) ) )
          & ( ( ( s @ A @ D )
              = ( s @ A @ C ) )
           => ( p @ ( s @ bool @ E ) ) )
          & ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ num ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ num ) ) @ B ) @ ( s @ A @ D ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1623]) ).

thf(2615,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_UNIV) ).

thf(18712,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2615]) ).

thf(2006,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ? [D: $i] :
        ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLOWERu_BOUNDu_FINITEu_SETu_REAL) ).

thf(15475,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ? [D: $i] :
        ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2006]) ).

thf(2169,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) )
    <=> ! [C: $i] :
          ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
         => ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
              & ! [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) )
                 => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWF) ).

thf(16310,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) )
       => ! [C: $i] :
            ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                & ! [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) )
                   => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
      & ( ! [C: $i] :
            ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
                & ! [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) )
                   => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2169]) ).

thf(2045,axiom,
    ( ( ! [A: $i,B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
   => ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWLOGu_LE) ).

thf(15635,plain,
    ( ( ! [A: $i,B: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
      & ! [A: $i,B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
   => ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ p0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2045]) ).

thf(2247,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_CMULu_SUBu_LDISTRIB) ).

thf(16706,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2247]) ).

thf(1620,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
      <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_EQu_0) ).

thf(13000,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1620]) ).

thf(1767,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
    ? [F: $i] :
      ( ! [G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ ( sum0 @ A @ C ) @ B ) @ F ) @ ( s @ ( sum0 @ A @ C ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ C ) ) @ inl ) @ ( s @ A @ G ) ) ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) )
      & ! [G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ ( sum0 @ A @ C ) @ B ) @ F ) @ ( s @ ( sum0 @ A @ C ) @ ( i @ ( s @ ( fun @ C @ ( sum0 @ A @ C ) ) @ inr ) @ ( s @ C @ G ) ) ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ E ) @ ( s @ C @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asumu_RECURSION) ).

thf(13944,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
    ? [F: $i] :
      ( ! [G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ ( sum0 @ A @ C ) @ B ) @ F ) @ ( s @ ( sum0 @ A @ C ) @ ( i @ ( s @ ( fun @ A @ ( sum0 @ A @ C ) ) @ inl ) @ ( s @ A @ G ) ) ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) )
      & ! [G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ ( sum0 @ A @ C ) @ B ) @ F ) @ ( s @ ( sum0 @ A @ C ) @ ( i @ ( s @ ( fun @ C @ ( sum0 @ A @ C ) ) @ inr ) @ ( s @ C @ G ) ) ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ E ) @ ( s @ C @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1767]) ).

thf(1527,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ numu_mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ int @ bool ) ) ) @ intu_mod ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_mod) ).

thf(12432,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ numu_mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ int @ bool ) ) ) @ intu_mod ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1527]) ).

thf(3174,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
    <=> ~ ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_NOTu_THM) ).

thf(21718,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ~ ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
      & ( ~ ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ! [C: $i] :
            ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3174]) ).

thf(2138,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMu_APPEND) ).

thf(16129,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2138]) ).

thf(332,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_TOTALu_LE) ).

thf(5216,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(2224,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
    <=> ? [E: $i,F: $i] :
          ( ! [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
                  = ( s @ A @ G ) ) ) )
          & ! [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) )
                  = ( s @ B @ G ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_Cu_BIJECTIONS) ).

thf(16599,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
       => ? [E: $i,F: $i] :
            ( ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
                    = ( s @ A @ G ) ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) )
                    = ( s @ B @ G ) ) ) ) ) )
      & ( ? [E: $i,F: $i] :
            ( ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
                    = ( s @ A @ G ) ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) )
                    = ( s @ B @ G ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2224]) ).

thf(934,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_conjunct0) ).

thf(8776,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[934]) ).

thf(2070,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LT2u_ODDu_EQ) ).

thf(15725,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2070]) ).

thf(3007,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOIDALu_VECTORu_ADD) ).

thf(20805,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3007]) ).

thf(326,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_NEGu_0) ).

thf(5197,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(3029,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i] :
              ( ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
         => ! [F: $i] :
              ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) )
            <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ G ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_UNCURRY) ).

thf(20924,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i] :
              ( ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
         => ! [F: $i] :
              ( ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) )
               => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ G ) ) ) ) ) ) )
              & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ G ) ) ) ) ) )
               => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3029]) ).

thf(2890,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_CLAUSESu_conjunct9) ).

thf(20201,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2890]) ).

thf(2693,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
     => ( ( s @ real @ A )
       != ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_IMPu_NE) ).

thf(19093,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
     => ( ( s @ real @ A )
       != ( s @ real @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2693]) ).

thf(333,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ f ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ f ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ f ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_CLAUSESu_NUMSEGu_conjunct1) ).

thf(5218,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ f ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ f ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ f ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).

thf(1799,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_num) ).

thf(14207,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1799]) ).

thf(2359,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i] :
              ( ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) )
            <=> ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ G ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_UNCURRY) ).

thf(17223,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i] :
              ( ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
         => ! [F: $i] :
              ( ( ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) )
               => ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ G ) ) ) ) ) ) )
              & ( ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ ( fun @ A @ ( fun @ B @ C ) ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ G ) ) ) ) ) )
               => ! [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2359]) ).

thf(1914,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_SQUAREu_REFL) ).

thf(14876,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1914]) ).

thf(1844,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ univ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_FUNSPACEu_UNIV) ).

thf(14453,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ univ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1844]) ).

thf(2279,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( ( s @ num @ A )
         != ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_LE) ).

thf(16863,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          & ( ( s @ num @ A )
           != ( s @ num @ B ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          & ( ( s @ num @ A )
           != ( s @ num @ B ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2279]) ).

thf(3144,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_ADDu_NUMSEG) ).

thf(21571,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3144]) ).

thf(1492,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h2 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALL2u_conjunct2) ).

thf(12135,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h2 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1492]) ).

thf(1590,axiom,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ A @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( list @ A ) ) ) @ replicate ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ A @ x ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ A @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( list @ A ) ) ) @ replicate ) @ ( s @ num @ n ) ) ) @ ( s @ A @ x ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREPLICATEu_conjunct1) ).

thf(12797,plain,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ A @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( list @ A ) ) ) @ replicate ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ A @ x ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ x ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ A @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( list @ A ) ) ) @ replicate ) @ ( s @ num @ n ) ) ) @ ( s @ A @ x ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1590]) ).

thf(2593,axiom,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTEu_ANTISYM) ).

thf(18602,plain,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2593]) ).

thf(948,axiom,
    ( ( ( p @ ( s @ bool @ p1 ) )
     => ( ( p @ ( s @ bool @ q ) )
       => ( p @ ( s @ bool @ r0 ) ) ) )
  <=> ( ( ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ q ) ) )
     => ( p @ ( s @ bool @ r0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMPu_IMP) ).

thf(8821,plain,
    ( ( ( ( p @ ( s @ bool @ p1 ) )
       => ( ( p @ ( s @ bool @ q ) )
         => ( p @ ( s @ bool @ r0 ) ) ) )
     => ( ( ( p @ ( s @ bool @ p1 ) )
          & ( p @ ( s @ bool @ q ) ) )
       => ( p @ ( s @ bool @ r0 ) ) ) )
    & ( ( ( ( p @ ( s @ bool @ p1 ) )
          & ( p @ ( s @ bool @ q ) ) )
       => ( p @ ( s @ bool @ r0 ) ) )
     => ( ( p @ ( s @ bool @ p1 ) )
       => ( ( p @ ( s @ bool @ q ) )
         => ( p @ ( s @ bool @ r0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[948]) ).

thf(488,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_NORMu_NEG) ).

thf(6151,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).

thf(1147,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
         => ! [D: $i] :
              ( ( s @ ( cart @ real @ A ) @ D )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_COMPONENTWISE) ).

thf(9899,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
         => ! [D: $i] :
              ( ( s @ ( cart @ real @ A ) @ D )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1147]) ).

thf(3010,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
        ? [E: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
          = ( s @ B @ D ) )
    <=> ! [D: $i] :
          ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) ) )
        <=> ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJECTIVEu_EXISTSu_THM) ).

thf(20815,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
          ? [E: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
            = ( s @ B @ D ) )
       => ! [D: $i] :
            ( ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) ) )
             => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) ) )
            & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) )
             => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) ) )
             => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) ) )
            & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) )
             => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
       => ! [D: $i] :
          ? [E: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ E ) ) )
            = ( s @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3010]) ).

thf(823,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_1u_conjunct5) ).

thf(8084,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[823]) ).

thf(403,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_0) ).

thf(5646,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).

thf(1083,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_OFu_NUMu_LE) ).

thf(9492,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1083]) ).

thf(514,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ f ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_1) ).

thf(6270,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ f ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).

thf(1728,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_RESTRICT) ).

thf(13666,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1728]) ).

thf(1132,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_RINV) ).

thf(9841,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1132]) ).

thf(505,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_RZERO) ).

thf(6223,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).

thf(2000,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ F ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ G ) ) ) )
        <=> ! [H: $i,I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ D @ bool ) ) @ geq ) @ ( s @ D @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ G ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) ) ) ) ) ) ) @ ( s @ D @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ F ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) ) ) ) ) ) ) ) )
     => ! [F: $i] :
          ( ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ F ) ) ) ) )
          = ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ F ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIREDu_ETAu_THMu_conjunct1) ).

thf(15432,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ F ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ G ) ) ) )
           => ! [H: $i,I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ D @ bool ) ) @ geq ) @ ( s @ D @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ G ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) ) ) ) ) ) ) @ ( s @ D @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ F ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) ) ) ) ) ) ) ) )
          & ( ! [H: $i,I: $i,J: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ D @ bool ) ) @ geq ) @ ( s @ D @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ G ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) ) ) ) ) ) ) @ ( s @ D @ ( i @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ F ) @ ( s @ ( prod @ A @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ B @ C ) @ ( prod @ A @ ( prod @ B @ C ) ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ F ) ) ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ G ) ) ) ) ) )
     => ! [F: $i] :
          ( ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ ( fun @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ bool ) ) @ E ) @ ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ F ) ) ) ) )
          = ( s @ ( fun @ ( prod @ A @ ( prod @ B @ C ) ) @ D ) @ F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2000]) ).

thf(596,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_LCANCELu_IMP) ).

thf(6783,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).

thf(97,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_EQu_0) ).

thf(3687,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(340,axiom,
    ! [A: $i] :
    ? [B: $i,C: $i] :
    ! [D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_UBOUND) ).

thf(5248,plain,
    ! [A: $i] :
    ? [B: $i,C: $i] :
    ! [D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).

thf(2090,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) @ ( s @ ( cart @ real @ A ) @ n ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ p1 ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ n ) ) ) @ ( s @ ( cart @ real @ A ) @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_ACu_conjunct1) ).

thf(15849,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) @ ( s @ ( cart @ real @ A ) @ n ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ p1 ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ n ) ) ) @ ( s @ ( cart @ real @ A ) @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2090]) ).

thf(2980,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSELECTu_AX) ).

thf(20660,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2980]) ).

thf(92,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_SQUARE) ).

thf(3674,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(1318,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_RMUL) ).

thf(10984,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1318]) ).

thf(580,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_RESTRICT) ).

thf(6704,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).

thf(2379,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_ADDu_CONG) ).

thf(17313,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ ( fun @ ( sum0 @ C @ D ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2379]) ).

thf(459,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_ONE) ).

thf(5976,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).

thf(2731,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_SUBu_COMPONENT) ).

thf(19288,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2731]) ).

thf(2442,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        | ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORu_EXISTSu_THM) ).

thf(17758,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2442]) ).

thf(2334,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ A ) ) ) ) )
    <=> ( ( s @ hreal @ A )
        = ( s @ hreal @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_LEu_ANTISYM) ).

thf(17102,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ A ) ) ) ) )
       => ( ( s @ hreal @ A )
          = ( s @ hreal @ B ) ) )
      & ( ( ( s @ hreal @ A )
          = ( s @ hreal @ B ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2334]) ).

thf(1524,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) )
    <=> ( p @ ( s @ bool @ f0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EQu_conjunct7) ).

thf(12409,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) )
       => ( p @ ( s @ bool @ f0 ) ) )
      & ( ( p @ ( s @ bool @ f0 ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1524]) ).

thf(2477,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_LE) ).

thf(17940,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2477]) ).

thf(1065,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ A )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ real @ B )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_INV) ).

thf(9366,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ A )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ real @ B )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1065]) ).

thf(2032,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_EVEN) ).

thf(15581,plain,
    ! [A: $i] :
      ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2032]) ).

thf(1180,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_GE0) ).

thf(10063,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1180]) ).

thf(2355,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
      & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
      & ! [F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_INJECTIVEu_IMAGEu_OFu_SUBSET) ).

thf(17191,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
      & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
      & ! [F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ G ) ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ G ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2355]) ).

thf(1138,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ D ) ) )
     => ! [F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_IMPu_INJECTIVEu_GEN) ).

thf(9871,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ D ) ) )
     => ! [F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ G ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1138]) ).

thf(1303,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      | ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SGNu_CASES) ).

thf(10803,plain,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      | ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1303]) ).

thf(673,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ h0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALLu_conjunct1) ).

thf(7200,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ h0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ h0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[673]) ).

thf(34,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_MONOu_LT) ).

thf(3325,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(985,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_INV2) ).

thf(9011,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[985]) ).

thf(200,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i,G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                <=> ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) )
             => ! [E: $i,F: $i] :
                  ( ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
                  = ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ union ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPOWERSETu_CLAUSESu_conjunct1) ).

thf(4304,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
              & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i,G: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                   => ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) )
                  & ( ? [H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
             => ! [E: $i,F: $i] :
                  ( ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
                  = ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ union ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(925,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_ALT) ).

thf(8676,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[925]) ).

thf(2997,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i] :
              ( ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ C ) ) ) @ ( s @ num @ D ) ) )
              = ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ! [C: $i] :
              ( ! [D: $i,E: $i,F: $i,G: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ F ) ) ) @ ( s @ num @ G ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ F ) @ ( s @ num @ G ) ) ) ) ) )
             => ! [D: $i,E: $i,F: $i,G: $i,H: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ H ) ) ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ H ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SWAPu_NUMSEG) ).

thf(20770,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i] :
              ( ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ C ) ) ) @ ( s @ num @ D ) ) )
              = ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ A ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) )
         => ! [C: $i] :
              ( ! [D: $i,E: $i,F: $i,G: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ F ) ) ) @ ( s @ num @ G ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ F ) @ ( s @ num @ G ) ) ) ) ) )
             => ! [D: $i,E: $i,F: $i,G: $i,H: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ H ) ) ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ real ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ H ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2997]) ).

thf(291,axiom,
    ( ( ( p @ ( s @ bool @ p1 ) )
      & ( p @ ( s @ bool @ q ) )
      & ( p @ ( s @ bool @ r0 ) ) )
  <=> ( ( p @ ( s @ bool @ p1 ) )
      & ( p @ ( s @ bool @ q ) )
      & ( p @ ( s @ bool @ r0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONJu_ACIu_conjunct1) ).

thf(4954,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(2972,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aGEu_REFL) ).

thf(20607,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2972]) ).

thf(703,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ARCHu_RDIVu_EQu_0) ).

thf(7339,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[703]) ).

thf(1888,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_NEG2) ).

thf(14675,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1888]) ).

thf(468,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_MULu_LID) ).

thf(6027,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).

thf(60,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ A ) ) )
     != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFACTu_NZ) ).

thf(3512,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ A ) ) )
     != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(1494,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_powu_th) ).

thf(12224,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1494]) ).

thf(2316,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_COMM) ).

thf(17006,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2316]) ).

thf(2684,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_UNIONS) ).

thf(19039,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2684]) ).

thf(2278,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_ASSOC) ).

thf(16860,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2278]) ).

thf(1117,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ F ) ) ) )
        <=> ! [G: $i,H: $i] :
            ? [I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ G ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_l_ ) @ ( s @ B @ E ) ) ) @ ( s @ B @ H ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ E ) ) ) )
            <=> ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) ) )
         => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_l_ ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_POINTWISE) ).

thf(9740,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ F ) ) ) )
           => ! [G: $i,H: $i] :
              ? [I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_l_ ) @ ( s @ B @ E ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_l_ ) @ ( s @ B @ E ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
          & ( ! [G: $i,H: $i] :
              ? [I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_l_ ) @ ( s @ B @ E ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_l_ ) @ ( s @ B @ E ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ E ) ) ) )
               => ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) ) )
              & ( ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ geq ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ D ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ E ) ) ) ) ) )
         => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_l_ ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ bool ) @ D ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1117]) ).

thf(2188,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ E ) ) ) ) )
    <=> ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_INu_IMAGE) ).

thf(16408,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ E ) ) ) ) )
       => ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) ) ) )
      & ( ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) ) )
       => ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2188]) ).

thf(2223,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_ADDu_LDISTRIB) ).

thf(16596,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2223]) ).

thf(281,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ A ) ) ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',atrealu_neg) ).

thf(4890,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ A ) ) ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(2893,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
        & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_ANDu_EXISTSu_THM) ).

thf(20208,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( p @ ( s @ bool @ B ) )
          & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ( p @ ( s @ bool @ B ) )
          & ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2893]) ).

thf(2998,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
        & ! [C: $i,D: $i] :
            ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSETu_PROVEu_CASES) ).

thf(20773,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
        & ! [C: $i,D: $i] :
            ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2998]) ).

thf(469,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_EQu_1) ).

thf(6029,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).

thf(2207,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_COMPOSEu_SUB) ).

thf(16510,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2207]) ).

thf(1877,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_DELETE) ).

thf(14628,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1877]) ).

thf(2778,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_INFINITE) ).

thf(19580,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2778]) ).

thf(548,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_LEu_0) ).

thf(6485,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).

thf(1145,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ y ) ) ) @ ( s @ real @ z ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ y ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ z ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_ACIu_conjunct2) ).

thf(9894,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ y ) ) ) @ ( s @ real @ z ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ y ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ x ) ) ) @ ( s @ real @ z ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1145]) ).

thf(2757,axiom,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ B ) ) ) ) ) )
     => ? [B: $i] :
          ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ B ) ) ) ) )
          & ! [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ D ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ D ) ) ) @ ( s @ nadd @ C ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_COMPLETE) ).

thf(19409,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ B ) ) ) ) ) )
     => ? [B: $i] :
          ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ B ) ) ) ) )
          & ! [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ D ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ D ) ) ) @ ( s @ nadd @ C ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2757]) ).

thf(1364,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_EQu_1) ).

thf(11361,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1364]) ).

thf(597,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
            <=> ? [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ H ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i,G: $i,H: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ F ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [E: $i,F: $i,G: $i,H: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                 => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ G ) ) ) @ ( s @ ( fun @ A @ num ) @ H ) ) ) ) )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ G ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ H ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_CASES) ).

thf(6785,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
          & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
               => ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
              & ( ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
         => ! [D: $i] :
              ( ! [E: $i,F: $i,G: $i,H: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ F ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [E: $i,F: $i,G: $i,H: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                 => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ G ) ) ) @ ( s @ ( fun @ A @ num ) @ H ) ) ) ) )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ G ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ H ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).

thf(2994,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
    <=> ? [E: $i] :
        ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
         => ? [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) )
                = ( s @ B @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_C) ).

thf(20734,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
       => ? [E: $i] :
          ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
           => ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) )
                  = ( s @ B @ F ) ) ) ) )
      & ( ? [E: $i] :
          ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
           => ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) )
                  = ( s @ B @ F ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2994]) ).

thf(1809,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
     => ( ( s @ int @ A )
       != ( s @ int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_IMPu_NE) ).

thf(14255,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
     => ( ( s @ int @ A )
       != ( s @ int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1809]) ).

thf(1871,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
    <=> ? [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
          & ( ( s @ ( fun @ A @ bool ) @ D )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_IMAGE) ).

thf(14599,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
       => ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
            & ( ( s @ ( fun @ A @ bool ) @ D )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
      & ( ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
            & ( ( s @ ( fun @ A @ bool ) @ D )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1871]) ).

thf(2430,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) )
        = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
    <=> ( ( ( s @ A @ B )
          = ( s @ A @ C ) )
        & ( ( s @ ( list @ A ) @ D )
          = ( s @ ( list @ A ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONSu_11) ).

thf(17664,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) )
          = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
       => ( ( ( s @ A @ B )
            = ( s @ A @ C ) )
          & ( ( s @ ( list @ A ) @ D )
            = ( s @ ( list @ A ) @ E ) ) ) )
      & ( ( ( ( s @ A @ B )
            = ( s @ A @ C ) )
          & ( ( s @ ( list @ A ) @ D )
            = ( s @ ( list @ A ) @ E ) ) )
       => ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) )
          = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2430]) ).

thf(1500,axiom,
    ! [A: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ hreal @ A ) ) )
      = ( s @ hreal @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_ADDu_LID) ).

thf(12251,plain,
    ! [A: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ hreal @ A ) ) )
      = ( s @ hreal @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1500]) ).

thf(667,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
     => ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ B @ F )
                = ( s @ B @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_INJ) ).

thf(7172,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
     => ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ B @ F )
                = ( s @ B @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[667]) ).

thf(162,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        <=> ? [D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ E ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                  | ( ( s @ A @ D )
                    = ( s @ A @ x ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERT) ).

thf(3981,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
           => ? [D: $i,E: $i] :
                ( ( ( p @ ( s @ bool @ E ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                    | ( ( s @ A @ D )
                      = ( s @ A @ x ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                    | ( ( s @ A @ D )
                      = ( s @ A @ x ) ) )
                 => ( p @ ( s @ bool @ E ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) )
          & ( ? [D: $i,E: $i] :
                ( ( ( p @ ( s @ bool @ E ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                    | ( ( s @ A @ D )
                      = ( s @ A @ x ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                    | ( ( s @ A @ D )
                      = ( s @ A @ x ) ) )
                 => ( p @ ( s @ bool @ E ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ A @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(626,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_MULTu_conjunct8) ).

thf(6962,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[626]) ).

thf(657,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_LNEG) ).

thf(7122,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[657]) ).

thf(3151,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ! [D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
             => ( ( s @ ( fun @ A @ bool ) @ D )
                = ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHP) ).

thf(21603,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ! [D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
             => ( ( s @ ( fun @ A @ bool ) @ D )
                = ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3151]) ).

thf(479,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAX) ).

thf(6083,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).

thf(1700,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ ( fun @ num @ num ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ C ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ ( fun @ num @ num ) @ C ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_ADDu_NUMSEG) ).

thf(13528,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ ( fun @ num @ num ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ C ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ ( fun @ num @ num ) @ C ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1700]) ).

thf(1934,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_MONOu_LEu_IMP) ).

thf(15048,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1934]) ).

thf(2129,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_NEGu_SUB) ).

thf(16094,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2129]) ).

thf(1603,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_OFFSET) ).

thf(12886,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1603]) ).

thf(1156,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_OFu_NUMu_MUL) ).

thf(9958,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1156]) ).

thf(593,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_UNION) ).

thf(6768,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).

thf(2665,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GTu_conjunct6) ).

thf(18935,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2665]) ).

thf(63,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ int @ num ) @ numu_ofu_int ) @ ( s @ int @ A ) ) ) ) )
        = ( s @ int @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_OFu_INT) ).

thf(3522,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ int @ num ) @ numu_ofu_int ) @ ( s @ int @ A ) ) ) ) )
        = ( s @ int @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(999,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_ADD) ).

thf(9054,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[999]) ).

thf(1612,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ! [B: $i] :
        ? [C: $i] :
          ( ( ( p @ ( s @ bool @ C ) )
          <=> ( ( s @ num @ B )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
          & ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) )
            = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ empty ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_LT) ).

thf(12922,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i] :
        ? [C: $i] :
          ( ( ( p @ ( s @ bool @ C ) )
           => ( ( s @ num @ B )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
          & ( ( ( s @ num @ B )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
           => ( p @ ( s @ bool @ C ) ) )
          & ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) )
            = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ empty ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1612]) ).

thf(1251,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) )
    <=> ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_MULTu_LCANCEL) ).

thf(10517,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ( ( ( s @ num @ A )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1251]) ).

thf(1989,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPSUBSETu_TRANS) ).

thf(15355,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1989]) ).

thf(1651,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ E ) ) ) ) ) ) ) )
         => ! [C: $i,D: $i,E: $i,F: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_PARTIALu_SUC) ).

thf(13268,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ E ) ) ) ) ) ) ) )
         => ! [C: $i,D: $i,E: $i,F: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ C ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ C ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1651]) ).

thf(409,axiom,
    ! [A: $i] :
      ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_FORALLu_POS) ).

thf(5688,plain,
    ! [A: $i] :
      ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) )
       => ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ B ) ) ) ) ) )
      & ( ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ B ) ) ) ) )
       => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ A ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).

thf(455,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_LEu_COMPONENTWISE) ).

thf(5959,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).

thf(976,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) ) @ ( s @ C @ b0 ) ) )
      = ( s @ C @ b0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITLIST2u_DEFu_conjunct0) ).

thf(8965,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( fun @ C @ C ) ) ) ) @ itlist2 ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ C ) ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) ) @ ( s @ C @ b0 ) ) )
      = ( s @ C @ b0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[976]) ).

thf(1676,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MIN) ).

thf(13371,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1676]) ).

thf(1401,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMPAIR) ).

thf(11665,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ numpair ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1401]) ).

thf(2321,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_INSERT) ).

thf(17038,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2321]) ).

thf(437,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asetu_ofu_listu_conjunct0) ).

thf(5890,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).

thf(2052,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ! [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_EQ) ).

thf(15657,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ! [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2052]) ).

thf(2654,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_DIFF) ).

thf(18891,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2654]) ).

thf(726,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_ADDu_SYMu_EQ) ).

thf(7490,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[726]) ).

thf(166,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
      <=> ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_CARD) ).

thf(4013,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        & ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(1469,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_NEG) ).

thf(12021,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1469]) ).

thf(655,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_LID) ).

thf(7117,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[655]) ).

thf(1540,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( ( s @ num @ E )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ) @ ( s @ num @ E ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_NSUMu_MODu_NUMSEG) ).

thf(12492,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( ( s @ num @ E )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ) @ ( s @ num @ E ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1540]) ).

thf(1897,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_UNION) ).

thf(14720,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1897]) ).

thf(1112,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) )
     => ( ! [E: $i] :
            ( ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ C ) @ E ) ) )
            = ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ C ) @ neutral ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) )
        & ! [E: $i,F: $i,G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
           => ( ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) )
              = ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ ( fun @ C @ C ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_CLAUSES) ).

thf(9673,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) )
     => ( ! [E: $i] :
            ( ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ C ) @ E ) ) )
            = ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ C ) @ neutral ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) )
        & ! [E: $i,F: $i,G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) )
           => ( ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) )
              = ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ ( fun @ C @ C ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ C @ C ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ E ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1112]) ).

thf(563,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_RSQRTu_LE) ).

thf(6542,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).

thf(2803,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ m ) ) ) @ ( s @ real @ n ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ n ) ) ) @ ( s @ real @ m ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_ACu_conjunct0) ).

thf(19695,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ m ) ) ) @ ( s @ real @ n ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ n ) ) ) @ ( s @ real @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2803]) ).

thf(2612,axiom,
    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ x ) ) ) )
  <=> ? [A: $i] :
        ( ( ( s @ real @ x )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) )
        | ( ( s @ real @ x )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aisu_int) ).

thf(18698,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ x ) ) ) )
     => ? [A: $i] :
          ( ( ( s @ real @ x )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) )
          | ( ( s @ real @ x )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) ) ) )
    & ( ? [A: $i] :
          ( ( ( s @ real @ x )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) )
          | ( ( s @ real @ x )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ x ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2612]) ).

thf(749,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ m ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ n ) ) ) @ ( s @ int @ p1 ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ n ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ m ) ) ) @ ( s @ int @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_ACu_conjunct2) ).

thf(7681,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ m ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ n ) ) ) @ ( s @ int @ p1 ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ n ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ m ) ) ) @ ( s @ int @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[749]) ).

thf(1753,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ? [D: $i] :
          ( ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
          & ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
             => ( ? [F: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ C ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ F ) ) )
                      = ( s @ A @ E ) ) )
                & ! [F: $i,G: $i] :
                    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ C ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ F ) ) )
                        = ( s @ A @ E ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ C ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ G ) ) )
                        = ( s @ A @ E ) ) )
                   => ( ( s @ num @ F )
                      = ( s @ num @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_INDEX) ).

thf(13852,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ? [D: $i] :
          ( ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
          & ! [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
             => ( ? [F: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ C ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ F ) ) )
                      = ( s @ A @ E ) ) )
                & ! [F: $i,G: $i] :
                    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ C ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ F ) ) )
                        = ( s @ A @ E ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ C ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ G ) ) )
                        = ( s @ A @ E ) ) )
                   => ( ( s @ num @ F )
                      = ( s @ num @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1753]) ).

thf(27,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ numu_gcd ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMu_GCD) ).

thf(3293,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ numu_gcd ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ int ) @ intu_gcd ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(2122,axiom,
    ! [A: $i] :
      ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) )
        & ! [B: $i,C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ( ! [B: $i,C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      <=> ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSITIVEu_STEPWISEu_LEu_EQ) ).

thf(16024,plain,
    ! [A: $i] :
      ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) )
        & ! [B: $i,C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ( ( ! [B: $i,C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
         => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) )
        & ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) )
         => ! [B: $i,C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2122]) ).

thf(2748,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAXu_MAX) ).

thf(19353,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2748]) ).

thf(2242,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) )
    <=> ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_PAIRu_THM) ).

thf(16674,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) )
       => ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) ) )
      & ( ? [D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) )
       => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2242]) ).

thf(1473,axiom,
    ! [A: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',atrealu_ofu_num) ).

thf(12048,plain,
    ! [A: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1473]) ).

thf(719,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_SUBSETu_LE) ).

thf(7447,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[719]) ).

thf(91,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_OFu_NUMu_MUL) ).

thf(3672,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(1722,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_TRIANGLEu_ADDu_HALF) ).

thf(13641,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1722]) ).

thf(1130,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
            & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
              = ( s @ real @ B ) ) ) )
     => ! [B: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ h_ ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asqrt) ).

thf(9826,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
              & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
                = ( s @ real @ B ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
              & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
                = ( s @ real @ B ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ h_ ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1130]) ).

thf(736,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) )
     => ( ( s @ int @ A )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_STILLNZ) ).

thf(7562,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) )
     => ( ( s @ int @ A )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[736]) ).

thf(2222,axiom,
    ( ( ( p @ ( s @ bool @ p1 ) )
      & ( p @ ( s @ bool @ p1 ) )
      & ( p @ ( s @ bool @ q ) ) )
  <=> ( ( p @ ( s @ bool @ p1 ) )
      & ( p @ ( s @ bool @ q ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONJu_ACIu_conjunct4) ).

thf(16589,plain,
    ( ( ( ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ q ) ) )
     => ( ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ q ) ) ) )
    & ( ( ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ q ) ) )
     => ( ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ q ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2222]) ).

thf(970,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_COMPONENT) ).

thf(8949,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[970]) ).

thf(938,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQUAREu_BOUNDu_LEMMA) ).

thf(8789,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[938]) ).

thf(1836,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
        <=> ? [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) )
     => ! [E: $i,F: $i] :
          ( ! [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ G ) ) ) ) )
        <=> ! [G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_INu_GSPECu_conjunct1) ).

thf(14406,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
           => ? [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) ) )
          & ( ? [H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
     => ! [E: $i,F: $i] :
          ( ( ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ G ) ) ) ) )
           => ! [G: $i,H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) ) )
          & ( ! [G: $i,H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ) )
           => ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ A ) ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ F ) ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1836]) ).

thf(2453,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',afl) ).

thf(17826,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2453]) ).

thf(474,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_DOTu_EQu_0u_conjunct0) ).

thf(6056,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ! [C: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).

thf(2403,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ n10 @ A ) @ C ) @ ( s @ n10 @ one ) ) )
      = ( s @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aoneu_RECURSION) ).

thf(17504,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ n10 @ A ) @ C ) @ ( s @ n10 @ one ) ) )
      = ( s @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2403]) ).

thf(1936,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_GE) ).

thf(15053,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1936]) ).

thf(952,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ! [B: $i] :
        ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ARCH) ).

thf(8836,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ! [B: $i] :
        ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[952]) ).

thf(670,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_LADD) ).

thf(7190,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[670]) ).

thf(2822,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_TOTAL) ).

thf(19789,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2822]) ).

thf(556,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_POW) ).

thf(6517,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).

thf(2142,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOIDALu_REALu_ADD) ).

thf(16148,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2142]) ).

thf(2747,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ C ) ) ) @ ( s @ num @ D ) ) ) )
    <=> ( ( ( s @ bool @ A )
          = ( s @ bool @ C ) )
        & ( ( s @ num @ B )
          = ( s @ num @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSUMu_INJ) ).

thf(19344,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ C ) ) ) @ ( s @ num @ D ) ) ) )
       => ( ( ( s @ bool @ A )
            = ( s @ bool @ C ) )
          & ( ( s @ num @ B )
            = ( s @ num @ D ) ) ) )
      & ( ( ( ( s @ bool @ A )
            = ( s @ bool @ C ) )
          & ( ( s @ num @ B )
            = ( s @ num @ D ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ num ) ) @ numsum ) @ ( s @ bool @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2747]) ).

thf(1859,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ char @ bool ) @ A ) @ ( s @ char @ ( i @ ( s @ ( fun @ bool @ char ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ char ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) ) ) @ ascii ) @ ( s @ bool @ B ) ) ) @ ( s @ bool @ C ) ) ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ char @ bool ) @ A ) @ ( s @ char @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',acharu_INDUCT) ).

thf(14530,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ char @ bool ) @ A ) @ ( s @ char @ ( i @ ( s @ ( fun @ bool @ char ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ char ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ char ) ) ) ) ) ) ) ) @ ascii ) @ ( s @ bool @ B ) ) ) @ ( s @ bool @ C ) ) ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ char @ bool ) @ A ) @ ( s @ char @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1859]) ).

thf(918,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_MUL) ).

thf(8657,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[918]) ).

thf(146,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_DOTu_EQu_0u_conjunct1) ).

thf(3921,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ! [C: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(2392,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ? [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
    <=> ? [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_FINITEu_SUBSETu_IMAGE) ).

thf(17400,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
       => ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) )
      & ( ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) )
       => ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2392]) ).

thf(2335,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_SYM) ).

thf(17112,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2335]) ).

thf(3155,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_UNIONS) ).

thf(21616,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3155]) ).

thf(2829,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_NEG2) ).

thf(19849,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) )
       => ( ( s @ real @ A )
          = ( s @ real @ B ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ B ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2829]) ).

thf(2130,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_NEG) ).

thf(16097,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2130]) ).

thf(2298,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_SYM) ).

thf(16948,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2298]) ).

thf(2709,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ int @ bool ) ) ) @ intu_mod ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_mod) ).

thf(19182,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( fun @ int @ bool ) ) ) @ intu_mod ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2709]) ).

thf(3032,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSsrnato_lequ_eqVlt) ).

thf(20936,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ B ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ B ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3032]) ).

thf(1687,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_DIFFS) ).

thf(13454,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) @ ( s @ num @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1687]) ).

thf(1297,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ D ) ) ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',atrealu_add) ).

thf(10788,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ D ) ) ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1297]) ).

thf(606,axiom,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
        = ( s @ A @ B ) )
      & ! [E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ E ) ) ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ A ) ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_RECURSION) ).

thf(6843,plain,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
        = ( s @ A @ B ) )
      & ! [E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ E ) ) ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ A ) ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).

thf(2529,axiom,
    ! [A: $i] :
    ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ARCH) ).

thf(18226,plain,
    ! [A: $i] :
    ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2529]) ).

thf(415,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_MUL) ).

thf(5763,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).

thf(3095,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( ( p @ ( s @ bool @ A ) )
            & ( p @ ( s @ bool @ B ) ) )
      <=> ( ~ ( p @ ( s @ bool @ A ) )
          | ~ ( p @ ( s @ bool @ B ) ) ) )
      & ( ~ ( ( p @ ( s @ bool @ A ) )
            | ( p @ ( s @ bool @ B ) ) )
      <=> ( ~ ( p @ ( s @ bool @ A ) )
          & ~ ( p @ ( s @ bool @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDEu_MORGANu_THM) ).

thf(21328,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( ( p @ ( s @ bool @ A ) )
            & ( p @ ( s @ bool @ B ) ) )
       => ( ~ ( p @ ( s @ bool @ A ) )
          | ~ ( p @ ( s @ bool @ B ) ) ) )
      & ( ( ~ ( p @ ( s @ bool @ A ) )
          | ~ ( p @ ( s @ bool @ B ) ) )
       => ~ ( ( p @ ( s @ bool @ A ) )
            & ( p @ ( s @ bool @ B ) ) ) )
      & ( ~ ( ( p @ ( s @ bool @ A ) )
            | ( p @ ( s @ bool @ B ) ) )
       => ( ~ ( p @ ( s @ bool @ A ) )
          & ~ ( p @ ( s @ bool @ B ) ) ) )
      & ( ( ~ ( p @ ( s @ bool @ A ) )
          & ~ ( p @ ( s @ bool @ B ) ) )
       => ~ ( ( p @ ( s @ bool @ A ) )
            | ( p @ ( s @ bool @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3095]) ).

thf(464,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) )
    <=> ( ( ( s @ num @ B )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        | ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_LTu_0) ).

thf(6008,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) )
       => ( ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          | ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          | ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).

thf(2658,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOIDALu_MUL) ).

thf(18908,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2658]) ).

thf(639,axiom,
    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) ) ) )
  <=> ! [A: $i] :
        ( ( ( ( s @ num @ a0 )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ A ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) )
            & ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_ELIMu_THM) ).

thf(6998,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) ) ) )
     => ! [A: $i] :
          ( ( ( ( s @ num @ a0 )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ A ) ) ) )
            | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) )
              & ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) )
    & ( ! [A: $i] :
          ( ( ( ( s @ num @ a0 )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ A ) ) ) )
            | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) )
              & ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[639]) ).

thf(839,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
    <=> ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
        | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_POSu_LT) ).

thf(8194,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
       => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
          | ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[839]) ).

thf(1496,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_0) ).

thf(12234,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1496]) ).

thf(2290,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_SUBu_RDISTRIB) ).

thf(16917,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2290]) ).

thf(204,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_LDIVu_EQ) ).

thf(4336,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(1244,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ nil ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALL2u_conjunct1) ).

thf(10471,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ nil ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1244]) ).

thf(1891,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ F ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
         => ! [E: $i] :
              ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrix ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amatrix) ).

thf(14688,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ F ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
         => ! [E: $i] :
              ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrix ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1891]) ).

thf(2343,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ D ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ! [B: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_REALu_INTERVALu_conjunct2) ).

thf(17151,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ D ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ D ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ! [B: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2343]) ).

thf(462,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ t ) ) ) @ ( s @ A @ D ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
        = ( s @ ( fun @ A @ bool ) @ univ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIVu_GSPEC) ).

thf(5991,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ t ) ) ) @ ( s @ A @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ t ) ) ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
        = ( s @ ( fun @ A @ bool ) @ univ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).

thf(2210,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_UNIQUE) ).

thf(16521,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2210]) ).

thf(103,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_INDUCTION) ).

thf(3737,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(2215,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LETu_TOTAL) ).

thf(16564,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2215]) ).

thf(994,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ m ) ) ) @ ( s @ real @ n ) ) ) ) ) @ ( s @ real @ p1 ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ m ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ n ) ) ) @ ( s @ real @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_ACu_conjunct1) ).

thf(9037,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ m ) ) ) @ ( s @ real @ n ) ) ) ) ) @ ( s @ real @ p1 ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ m ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ n ) ) ) @ ( s @ real @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[994]) ).

thf(2667,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ E ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
               => ( ? [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ E ) @ ( s @ B @ H ) ) )
                        = ( s @ B @ G ) ) )
                  & ! [H: $i,I: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ E ) @ ( s @ B @ H ) ) )
                          = ( s @ B @ G ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ E ) @ ( s @ B @ I ) ) )
                          = ( s @ B @ G ) ) )
                     => ( ( s @ B @ H )
                        = ( s @ B @ I ) ) ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ B ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ B ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ B ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_BIJECTION) ).

thf(18940,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ E ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
            & ! [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
               => ( ? [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ E ) @ ( s @ B @ H ) ) )
                        = ( s @ B @ G ) ) )
                  & ! [H: $i,I: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ E ) @ ( s @ B @ H ) ) )
                          = ( s @ B @ G ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                        & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ E ) @ ( s @ B @ I ) ) )
                          = ( s @ B @ G ) ) )
                     => ( ( s @ B @ H )
                        = ( s @ B @ I ) ) ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ B ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ B ) @ ( fun @ B @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ B ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2667]) ).

thf(734,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAXu_MIN) ).

thf(7554,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[734]) ).

thf(1274,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_ACIu_conjunct2) ).

thf(10653,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1274]) ).

thf(250,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_INCLu_EXCL) ).

thf(4658,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(2797,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
         => ( ( s @ ( fun @ B @ bool ) @ D )
            = ( s @ ( fun @ B @ bool ) @ E ) ) )
    <=> ! [D: $i,E: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
         => ( ( s @ B @ D )
            = ( s @ B @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJECTIVEu_IMAGE) ).

thf(19660,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i,E: $i] :
            ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( ( s @ ( fun @ B @ bool ) @ D )
              = ( s @ ( fun @ B @ bool ) @ E ) ) )
       => ! [D: $i,E: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
           => ( ( s @ B @ D )
              = ( s @ B @ E ) ) ) )
      & ( ! [D: $i,E: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
           => ( ( s @ B @ D )
              = ( s @ B @ E ) ) )
       => ! [D: $i,E: $i] :
            ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( ( s @ ( fun @ B @ bool ) @ D )
              = ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2797]) ).

thf(2271,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_ADDu_RCANCEL) ).

thf(16828,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2271]) ).

thf(2686,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_MULu_LTu_INFINITE) ).

thf(19050,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2686]) ).

thf(2084,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_IMPu_COUNTABLE) ).

thf(15817,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2084]) ).

thf(2117,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_SUBSET) ).

thf(16001,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2117]) ).

thf(1301,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ E ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_LEu_NUMSEG) ).

thf(10797,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ B ) @ ( s @ num @ E ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1301]) ).

thf(989,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( list @ A ) @ B )
        = ( s @ ( list @ A ) @ nil ) )
      | ? [C: $i,D: $i] :
          ( ( s @ ( list @ A ) @ B )
          = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alistu_CASES) ).

thf(9022,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( list @ A ) @ B )
        = ( s @ ( list @ A ) @ nil ) )
      | ? [C: $i,D: $i] :
          ( ( s @ ( list @ A ) @ B )
          = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[989]) ).

thf(337,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
        = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
        = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_NEGu_EQu_0) ).

thf(5229,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
          = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
          = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
          = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
          = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).

thf(1747,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ADDu_conjunct0) ).

thf(13828,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1747]) ).

thf(1511,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOMPONENTu_LEu_NORM) ).

thf(12289,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1511]) ).

thf(808,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_BOUNDu_LTu_GEN) ).

thf(8003,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[808]) ).

thf(2024,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_TRIANGLEu_LE) ).

thf(15551,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2024]) ).

thf(1538,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LTu_conjunct4) ).

thf(12487,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1538]) ).

thf(665,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_ADD1) ).

thf(7166,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[665]) ).

thf(33,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) )
    <=> ( ( ( s @ int @ A )
          = ( s @ int @ B ) )
        | ( ( s @ int @ C )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EQu_MULu_RCANCEL) ).

thf(3315,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) )
       => ( ( ( s @ int @ A )
            = ( s @ int @ B ) )
          | ( ( s @ int @ C )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      & ( ( ( ( s @ int @ A )
            = ( s @ int @ B ) )
          | ( ( s @ int @ C )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(1614,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_NUMSEGu_LE) ).

thf(12947,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1614]) ).

thf(1833,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i,E: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
           => ( ( s @ B @ D )
              = ( s @ B @ E ) ) )
        & ! [D: $i] :
          ? [E: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
            = ( s @ A @ D ) ) )
    <=> ? [D: $i] :
          ( ! [E: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) ) ) )
              = ( s @ A @ E ) )
          & ! [E: $i] :
              ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) )
              = ( s @ B @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIJECTIVEu_LEFTu_RIGHTu_INVERSE) ).

thf(14374,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ! [D: $i,E: $i] :
              ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
             => ( ( s @ B @ D )
                = ( s @ B @ E ) ) )
          & ! [D: $i] :
            ? [E: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
              = ( s @ A @ D ) ) )
       => ? [D: $i] :
            ( ! [E: $i] :
                ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) ) ) )
                = ( s @ A @ E ) )
            & ! [E: $i] :
                ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) )
                = ( s @ B @ E ) ) ) )
      & ( ? [D: $i] :
            ( ! [E: $i] :
                ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) ) ) )
                = ( s @ A @ E ) )
            & ! [E: $i] :
                ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) )
                = ( s @ B @ E ) ) )
       => ( ! [D: $i,E: $i] :
              ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
             => ( ( s @ B @ D )
                = ( s @ B @ E ) ) )
          & ! [D: $i] :
            ? [E: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
              = ( s @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1833]) ).

thf(800,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_sub) ).

thf(7942,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).

thf(350,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ D ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) ) )
    <=> ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',atrealu_eq) ).

thf(5294,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ D ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) ) )
       => ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
          = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) )
      & ( ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
          = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ D ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).

thf(1442,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ append ) @ ( s @ ( list @ B ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_APPEND) ).

thf(11909,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ append ) @ ( s @ ( list @ B ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1442]) ).

thf(1134,axiom,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
      <=> ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_ZERO) ).

thf(9847,plain,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
       => ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ B ) ) )
      & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1134]) ).

thf(447,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EXPu_conjunct8) ).

thf(5927,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).

thf(3004,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_ge ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_GE) ).

thf(20796,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_ge ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3004]) ).

thf(2587,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_POW) ).

thf(18561,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2587]) ).

thf(1879,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_REFL) ).

thf(14635,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1879]) ).

thf(1035,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSINGu_SUBSET) ).

thf(9253,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1035]) ).

thf(1771,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i,I: $i,J: $i,K: $i,L: $i,M: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ B @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ B ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ E @ A ) @ J ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ K ) ) ) @ ( s @ C @ L ) ) ) @ ( s @ A @ M ) ) ) )
        <=> ? [N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ u_guardedu_pattern ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ G ) @ ( s @ D @ N ) ) ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ I ) @ ( s @ C @ L ) ) ) @ ( s @ D @ N ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ J ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ K ) @ ( s @ C @ L ) ) ) @ ( s @ D @ N ) ) ) ) ) ) ) @ ( s @ A @ M ) ) ) ) ) ) )
     => ! [G: $i] :
          ( ! [H: $i,I: $i,J: $i,K: $i,L: $i,M: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ D @ B ) @ H ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ E @ A ) @ J ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ K ) ) ) @ ( s @ C @ L ) ) ) @ ( s @ B @ M ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ B @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ B ) @ H ) ) ) @ ( s @ B @ M ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ E @ A ) @ J ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ K ) ) ) @ ( s @ C @ L ) ) ) )
         => ! [H: $i] :
              ( ! [I: $i,J: $i,K: $i,L: $i,M: $i,N: $i] :
                  ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ E @ A ) @ L ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ C @ N ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ A ) ) @ u_match ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ I ) @ ( s @ C @ N ) ) ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ E @ A ) @ L ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ C @ N ) ) ) ) ) )
             => ! [I: $i] :
                  ( ! [J: $i,K: $i,L: $i,M: $i,N: $i] :
                      ( ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) ) @ I ) @ ( s @ ( fun @ C @ B ) @ J ) ) ) @ ( s @ ( fun @ D @ B ) @ K ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ ( fun @ E @ A ) @ N ) ) )
                      = ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ J ) ) ) @ ( s @ ( fun @ D @ B ) @ K ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ N ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ M ) ) ) )
                 => ! [J: $i,K: $i,L: $i,M: $i,N: $i,O: $i,P: $i] :
                      ( ( ! [Q: $i,R: $i,S: $i,T: $i] :
                            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ Q ) ) ) @ ( s @ C @ R ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ S ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ R ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ O ) @ ( s @ C @ R ) ) ) @ ( s @ D @ S ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ T ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ R ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ O ) @ ( s @ C @ R ) ) ) @ ( s @ D @ T ) ) ) ) )
                           => ( ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ P ) @ ( s @ C @ R ) ) ) @ ( s @ D @ S ) ) )
                              = ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ P ) @ ( s @ C @ R ) ) ) @ ( s @ D @ T ) ) ) ) )
                        & ! [Q: $i,R: $i,S: $i] :
                            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ Q ) ) ) @ ( s @ C @ R ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ O ) @ ( s @ C @ R ) ) ) @ ( s @ D @ S ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ S ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ R ) ) ) ) )
                           => ! [T: $i] :
                                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) @ ( s @ E @ T ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ P ) @ ( s @ C @ R ) ) ) @ ( s @ D @ S ) ) ) ) ) )
                               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) @ ( s @ E @ T ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ C @ E ) @ L ) @ ( s @ C @ R ) ) ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ C @ E ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) ) @ I ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ ( fun @ D @ B ) @ N ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ P ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPERADMISSIBLEu_MATCHu_GUARDEDu_PATTERN) ).

thf(13967,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i,I: $i,J: $i,K: $i,L: $i,M: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ B @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ B ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ E @ A ) @ J ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ K ) ) ) @ ( s @ C @ L ) ) ) @ ( s @ A @ M ) ) ) )
           => ? [N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ u_guardedu_pattern ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ G ) @ ( s @ D @ N ) ) ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ I ) @ ( s @ C @ L ) ) ) @ ( s @ D @ N ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ J ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ K ) @ ( s @ C @ L ) ) ) @ ( s @ D @ N ) ) ) ) ) ) ) @ ( s @ A @ M ) ) ) ) ) ) )
          & ( ? [N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ u_guardedu_pattern ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ G ) @ ( s @ D @ N ) ) ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ I ) @ ( s @ C @ L ) ) ) @ ( s @ D @ N ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ J ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ K ) @ ( s @ C @ L ) ) ) @ ( s @ D @ N ) ) ) ) ) ) ) @ ( s @ A @ M ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ B @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ B ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ E @ A ) @ J ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ K ) ) ) @ ( s @ C @ L ) ) ) @ ( s @ A @ M ) ) ) ) ) )
     => ! [G: $i] :
          ( ! [H: $i,I: $i,J: $i,K: $i,L: $i,M: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ D @ B ) @ H ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ E @ A ) @ J ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ K ) ) ) @ ( s @ C @ L ) ) ) @ ( s @ B @ M ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ B @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ B ) @ H ) ) ) @ ( s @ B @ M ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ E @ A ) @ J ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ K ) ) ) @ ( s @ C @ L ) ) ) )
         => ! [H: $i] :
              ( ! [I: $i,J: $i,K: $i,L: $i,M: $i,N: $i] :
                  ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ E @ A ) @ L ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ C @ N ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ A ) ) @ u_match ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ I ) @ ( s @ C @ N ) ) ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ E @ A ) @ L ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ C @ N ) ) ) ) ) )
             => ! [I: $i] :
                  ( ! [J: $i,K: $i,L: $i,M: $i,N: $i] :
                      ( ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) ) @ I ) @ ( s @ ( fun @ C @ B ) @ J ) ) ) @ ( s @ ( fun @ D @ B ) @ K ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ M ) ) ) @ ( s @ ( fun @ E @ A ) @ N ) ) )
                      = ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ J ) ) ) @ ( s @ ( fun @ D @ B ) @ K ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ N ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ M ) ) ) )
                 => ! [J: $i,K: $i,L: $i,M: $i,N: $i,O: $i,P: $i] :
                      ( ( ! [Q: $i,R: $i,S: $i,T: $i] :
                            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ Q ) ) ) @ ( s @ C @ R ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ S ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ R ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ O ) @ ( s @ C @ R ) ) ) @ ( s @ D @ S ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ T ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ R ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ O ) @ ( s @ C @ R ) ) ) @ ( s @ D @ T ) ) ) ) )
                           => ( ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ P ) @ ( s @ C @ R ) ) ) @ ( s @ D @ S ) ) )
                              = ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ P ) @ ( s @ C @ R ) ) ) @ ( s @ D @ T ) ) ) ) )
                        & ! [Q: $i,R: $i,S: $i] :
                            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ Q ) ) ) @ ( s @ C @ R ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ O ) @ ( s @ C @ R ) ) ) @ ( s @ D @ S ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ S ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ R ) ) ) ) )
                           => ! [T: $i] :
                                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) @ ( s @ E @ T ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ P ) @ ( s @ C @ R ) ) ) @ ( s @ D @ S ) ) ) ) ) )
                               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) @ ( s @ E @ T ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ C @ E ) @ L ) @ ( s @ C @ R ) ) ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ C @ E ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) ) @ I ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ ( fun @ D @ B ) @ N ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ P ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1771]) ).

thf(2702,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINSEGu_INSEG) ).

thf(19138,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2702]) ).

thf(1167,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_DISCRETE) ).

thf(10001,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1167]) ).

thf(1379,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ x ) ) )
    = ( s @ int @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAXu_ACIu_conjunct3) ).

thf(11460,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ x ) ) )
    = ( s @ int @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1379]) ).

thf(2826,axiom,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
           => ( ( s @ A @ C )
              = ( s @ A @ D ) ) ) )
    <=> ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          & ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
             => ( ( s @ A @ D )
                = ( s @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_UNIQUE) ).

thf(19821,plain,
    ! [A: $i,B: $i] :
      ( ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) ) )
       => ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
            & ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
               => ( ( s @ A @ D )
                  = ( s @ A @ C ) ) ) ) )
      & ( ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
            & ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
               => ( ( s @ A @ D )
                  = ( s @ A @ C ) ) ) )
       => ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2826]) ).

thf(1392,axiom,
    ( ( ( ( p @ ( s @ bool @ a ) )
       => ( p @ ( s @ bool @ b ) ) )
      & ( ( p @ ( s @ bool @ c0 ) )
       => ( p @ ( s @ bool @ d0 ) ) ) )
   => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ b0 ) ) ) @ ( s @ bool @ a ) ) ) @ ( s @ bool @ c0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ b0 ) ) ) @ ( s @ bool @ b ) ) ) @ ( s @ bool @ d0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOu_COND) ).

thf(11573,plain,
    ( ( ( ( p @ ( s @ bool @ a ) )
       => ( p @ ( s @ bool @ b ) ) )
      & ( ( p @ ( s @ bool @ c0 ) )
       => ( p @ ( s @ bool @ d0 ) ) ) )
   => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ b0 ) ) ) @ ( s @ bool @ a ) ) ) @ ( s @ bool @ c0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ b0 ) ) ) @ ( s @ bool @ b ) ) ) @ ( s @ bool @ d0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1392]) ).

thf(2647,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ A ) )
        | ( p @ ( s @ bool @ B ) )
        | ( p @ ( s @ bool @ C ) ) )
    <=> ( ( p @ ( s @ bool @ A ) )
        | ( p @ ( s @ bool @ B ) )
        | ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJu_ASSOC) ).

thf(18848,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2647]) ).

thf(341,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POW2u_ABS) ).

thf(5250,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(229,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_SIGN2) ).

thf(4490,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(2460,axiom,
    ! [A: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ( ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
        & ! [B: $i] :
            ( ! [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINF) ).

thf(17850,plain,
    ! [A: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ? [B: $i] :
          ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ( ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
        & ! [B: $i] :
            ( ! [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2460]) ).

thf(2597,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEMPTYu_DIFF) ).

thf(18622,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2597]) ).

thf(2723,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_SUBu_RADD) ).

thf(19261,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2723]) ).

thf(1414,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ B ) @ fst ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
      = ( s @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFST) ).

thf(11753,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ B @ A ) @ B ) @ fst ) @ ( s @ ( prod @ B @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ B @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( prod @ B @ A ) ) ) @ c_ ) @ ( s @ B @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
      = ( s @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1414]) ).

thf(364,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ B @ num ) ) @ C ) @ ( s @ ( fun @ B @ num ) @ D ) ) ) @ ( s @ B @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ D ) @ ( s @ B @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) )
            <=> ? [I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ J ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) )
             => ! [F: $i] :
                  ( ! [G: $i,H: $i,I: $i,J: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ B @ I ) ) ) @ ( s @ A @ J ) ) ) )
                    <=> ? [K: $i,L: $i] :
                          ( ( ( p @ ( s @ bool @ L ) )
                          <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ K ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ J ) ) ) @ ( s @ bool @ L ) ) ) @ ( s @ A @ K ) ) ) ) ) )
                 => ! [G: $i,H: $i,I: $i,J: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) )
                        & ! [K: $i] :
                            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ K ) ) ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) )
                           => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ K ) ) ) ) ) ) )
                              = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ J ) @ ( s @ B @ K ) ) ) ) ) )
                     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) ) )
                        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ B @ num ) ) @ C ) @ ( s @ ( fun @ B @ num ) @ J ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_MULTICOUNTu_GEN) ).

thf(5388,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ B @ num ) ) @ C ) @ ( s @ ( fun @ B @ num ) @ D ) ) ) @ ( s @ B @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ D ) @ ( s @ B @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) )
               => ? [I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) ) )
              & ( ? [I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ B @ I ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) ) ) ) ) )
             => ! [F: $i] :
                  ( ! [G: $i,H: $i,I: $i,J: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ B @ I ) ) ) @ ( s @ A @ J ) ) ) )
                       => ? [K: $i,L: $i] :
                            ( ( ( p @ ( s @ bool @ L ) )
                             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ K ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                            & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ K ) ) ) @ ( s @ B @ I ) ) ) ) )
                             => ( p @ ( s @ bool @ L ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ J ) ) ) @ ( s @ bool @ L ) ) ) @ ( s @ A @ K ) ) ) ) ) )
                      & ( ? [K: $i,L: $i] :
                            ( ( ( p @ ( s @ bool @ L ) )
                             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ K ) ) ) @ ( s @ B @ I ) ) ) ) ) )
                            & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ K ) ) ) @ ( s @ B @ I ) ) ) ) )
                             => ( p @ ( s @ bool @ L ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ J ) ) ) @ ( s @ bool @ L ) ) ) @ ( s @ A @ K ) ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ B @ I ) ) ) @ ( s @ A @ J ) ) ) ) ) )
                 => ! [G: $i,H: $i,I: $i,J: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) )
                        & ! [K: $i] :
                            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ K ) ) ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) )
                           => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ K ) ) ) ) ) ) )
                              = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ J ) @ ( s @ B @ K ) ) ) ) ) )
                     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) ) )
                        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ B @ num ) ) @ C ) @ ( s @ ( fun @ B @ num ) @ J ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).

thf(2123,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_ADD) ).

thf(16032,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2123]) ).

thf(902,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_POWu_2) ).

thf(8537,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[902]) ).

thf(2483,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_ANTISYMu_EQ) ).

thf(17968,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ C ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ C ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2483]) ).

thf(1388,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIT0u_DEFu_conjunct0) ).

thf(11562,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1388]) ).

thf(192,axiom,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ c ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ d ) ) ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ d ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ c ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ b0 ) ) ) ) ) ) )
   => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ c ) ) ) @ ( s @ num @ d ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_EQu_EXCLUSION) ).

thf(4272,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ b0 ) ) ) @ ( s @ num @ c ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ d ) ) ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ d ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ c ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ b0 ) ) ) ) ) ) )
   => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ a0 ) ) ) @ ( s @ num @ b0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ c ) ) ) @ ( s @ num @ d ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(1734,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_MULTu_conjunct9) ).

thf(13698,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1734]) ).

thf(3170,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ B @ A ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( sum0 @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ B @ A ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_ADDu_SYM) ).

thf(21696,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ B @ A ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( sum0 @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ B @ A ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3170]) ).

thf(2342,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDELETEu_NONu_ELEMENT) ).

thf(17144,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) )
          = ( s @ ( fun @ A @ bool ) @ C ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) )
          = ( s @ ( fun @ A @ bool ) @ C ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2342]) ).

thf(2060,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ B ) ) ) ) )
     => ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ B ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOu_EXISTS) ).

thf(15693,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ B ) ) ) ) )
     => ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ B ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2060]) ).

thf(1506,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLENGTHu_conjunct1) ).

thf(12271,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1506]) ).

thf(3136,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_COMPONENT) ).

thf(21535,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3136]) ).

thf(2368,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_ORu_ODD) ).

thf(17276,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2368]) ).

thf(2301,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_FINITEu_CONG) ).

thf(16955,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2301]) ).

thf(2871,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_LT) ).

thf(20109,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2871]) ).

thf(1864,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPPORTu_SUBSET) ).

thf(14557,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1864]) ).

thf(2849,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ A ) )
    <=> ( ( s @ bool @ ( i @ ( s @ ( fun @ n10 @ bool ) @ oneu_rep ) @ ( s @ n10 @ ( i @ ( s @ ( fun @ bool @ n10 ) @ oneu_abs ) @ ( s @ bool @ A ) ) ) ) )
        = ( s @ bool @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aoneu_tydefu_conjunct1) ).

thf(19954,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ A ) )
       => ( ( s @ bool @ ( i @ ( s @ ( fun @ n10 @ bool ) @ oneu_rep ) @ ( s @ n10 @ ( i @ ( s @ ( fun @ bool @ n10 ) @ oneu_abs ) @ ( s @ bool @ A ) ) ) ) )
          = ( s @ bool @ A ) ) )
      & ( ( ( s @ bool @ ( i @ ( s @ ( fun @ n10 @ bool ) @ oneu_rep ) @ ( s @ n10 @ ( i @ ( s @ ( fun @ bool @ n10 ) @ oneu_abs ) @ ( s @ bool @ A ) ) ) ) )
          = ( s @ bool @ A ) )
       => ( p @ ( s @ bool @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2849]) ).

thf(2668,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIFFu_INSERT) ).

thf(18957,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2668]) ).

thf(2948,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_LDISTRIB) ).

thf(20496,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2948]) ).

thf(373,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_HALFu_conjunct1) ).

thf(5484,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).

thf(601,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ZEROu_conjunct0) ).

thf(6826,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).

thf(53,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_ADD) ).

thf(3462,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(1657,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_ADDL) ).

thf(13285,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1657]) ).

thf(2536,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ B @ A ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_MULu_SYM) ).

thf(18290,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ B @ A ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2536]) ).

thf(1002,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_LNEG) ).

thf(9061,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1002]) ).

thf(1410,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
                = ( s @ A @ G ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) )
                = ( s @ B @ G ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ n ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ n ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIJECTIONSu_HASu_SIZE) ).

thf(11726,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
                = ( s @ A @ G ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) )
                = ( s @ B @ G ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ n ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ n ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1410]) ).

thf(2185,axiom,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ butlast ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( list @ A ) @ nil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBUTLASTu_conjunct0) ).

thf(16396,plain,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ butlast ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( list @ A ) @ nil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2185]) ).

thf(1397,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            <=> ? [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ H ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_FINITEu_SUBSETS) ).

thf(11594,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
               => ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
              & ( ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1397]) ).

thf(1675,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
     != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIMINDEXu_NONZERO) ).

thf(13367,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
     != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1675]) ).

thf(2594,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_ASSOC) ).

thf(18605,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2594]) ).

thf(2493,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) )
    <=> ? [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ bool ) ) @ mem ) @ ( s @ B @ F ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
          & ( ( s @ A @ D )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMu_MAP) ).

thf(18009,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) )
       => ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ bool ) ) @ mem ) @ ( s @ B @ F ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
            & ( ( s @ A @ D )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) ) )
      & ( ? [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ bool ) ) @ mem ) @ ( s @ B @ F ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
            & ( ( s @ A @ D )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2493]) ).

thf(2435,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_RNEG) ).

thf(17717,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2435]) ).

thf(1743,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ARCHu_MULT) ).

thf(13813,plain,
    ! [A: $i,B: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1743]) ).

thf(2653,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ E ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ delete ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ B @ E ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_DELETEu_INJ) ).

thf(18886,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
         => ( ( s @ B @ F )
            = ( s @ B @ E ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ delete ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ B @ E ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2653]) ).

thf(1627,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ F ) ) ) )
        & ! [G: $i,H: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ H ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ H ) ) ) @ ( s @ real @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ G ) ) ) ) )
           => ? [I: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ I ) ) ) )
                & ! [J: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ J ) ) ) @ ( s @ real @ H ) ) ) ) ) ) ) @ ( s @ real @ I ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ J ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ H ) ) ) ) ) ) ) ) ) @ ( s @ real @ G ) ) ) ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ E ) ) ) )
           => ? [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ H ) ) ) )
                & ! [I: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ I ) ) ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) ) ) ) ) @ ( s @ real @ H ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ I ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ E ) ) ) ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ F ) ) ) )
           => ? [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ H ) ) ) )
                & ! [I: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ I ) ) ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) ) ) ) ) @ ( s @ real @ H ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ I ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ F ) ) ) ) ) ) )
        & ~ ? [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ G ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ G ) ) ) @ ( s @ real @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ G ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ G ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ F ) ) ) ) ) )
     => ? [G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ G ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ G ) ) ) @ ( s @ real @ D ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ G ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ E ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ G ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONNECTEDu_REALu_LEMMA) ).

thf(13042,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ F ) ) ) )
        & ! [G: $i,H: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ H ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ H ) ) ) @ ( s @ real @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ G ) ) ) ) )
           => ? [I: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ I ) ) ) )
                & ! [J: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ J ) ) ) @ ( s @ real @ H ) ) ) ) ) ) ) @ ( s @ real @ I ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ J ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ H ) ) ) ) ) ) ) ) ) @ ( s @ real @ G ) ) ) ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ E ) ) ) )
           => ? [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ H ) ) ) )
                & ! [I: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ I ) ) ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) ) ) ) ) @ ( s @ real @ H ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ I ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ E ) ) ) ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ F ) ) ) )
           => ? [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ H ) ) ) )
                & ! [I: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ I ) ) ) @ ( s @ ( cart @ real @ A ) @ G ) ) ) ) ) ) ) @ ( s @ real @ H ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ I ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ F ) ) ) ) ) ) )
        & ~ ? [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ G ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ G ) ) ) @ ( s @ real @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ G ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ G ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ F ) ) ) ) ) )
     => ? [G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ G ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ G ) ) ) @ ( s @ real @ D ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ G ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ E ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ real @ ( cart @ real @ A ) ) @ B ) @ ( s @ real @ G ) ) ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1627]) ).

thf(803,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) ) ) )
    <=> ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( cart @ A @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ A ) @ D ) ) )
        = ( s @ ( cart @ A @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLAMBDAu_UNIQUE) ).

thf(7967,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) ) ) )
       => ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( cart @ A @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ A ) @ D ) ) )
          = ( s @ ( cart @ A @ B ) @ C ) ) )
      & ( ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( cart @ A @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ A ) @ D ) ) )
          = ( s @ ( cart @ A @ B ) @ C ) )
       => ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) @ ( s @ num @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[803]) ).

thf(1761,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ B )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ div0 ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_DIVISION) ).

thf(13916,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ B )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ div0 ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1761]) ).

thf(136,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ? [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) )
                    = ( s @ A @ E ) ) )
              & ! [F: $i,G: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) )
                      = ( s @ A @ E ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ G ) ) )
                      = ( s @ A @ E ) ) )
                 => ( ( s @ A @ F )
                    = ( s @ A @ G ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ num ) ) ) @ o ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_BIJECTION) ).

thf(3872,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ? [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) )
                    = ( s @ A @ E ) ) )
              & ! [F: $i,G: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) )
                      = ( s @ A @ E ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ G ) ) )
                      = ( s @ A @ E ) ) )
                 => ( ( s @ A @ F )
                    = ( s @ A @ G ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ num ) ) ) @ o ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(211,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ D ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',atrealu_le) ).

thf(4377,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ D ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(1350,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_LDIVu_EQ) ).

thf(11252,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1350]) ).

thf(857,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( list @ B ) @ num ) @ length ) @ ( s @ ( list @ B ) @ D ) ) ) )
     => ( ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ B ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ ( prod @ A @ B ) @ B ) @ snd ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) ) ) )
        = ( s @ ( list @ B ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_SNDu_ZIP) ).

thf(8295,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( list @ B ) @ num ) @ length ) @ ( s @ ( list @ B ) @ D ) ) ) )
     => ( ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ B ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ ( prod @ A @ B ) @ B ) @ snd ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) ) ) )
        = ( s @ ( list @ B ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[857]) ).

thf(1154,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
          <=> ( ( s @ num @ B )
              = ( s @ num @ C ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_BASISu_BASIS) ).

thf(9948,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
           => ( ( s @ num @ B )
              = ( s @ num @ C ) ) )
          & ( ( ( s @ num @ B )
              = ( s @ num @ C ) )
           => ( p @ ( s @ bool @ D ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1154]) ).

thf(946,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ C ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVMODu_UNIQ) ).

thf(8813,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ C ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[946]) ).

thf(1957,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LE2u_ODDu_EQ) ).

thf(15163,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1957]) ).

thf(1206,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
        & ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_BOUNDu_LT) ).

thf(10228,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
        & ? [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1206]) ).

thf(1750,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ nil ) ) )
      = ( s @ ( list @ ( prod @ A @ B ) ) @ nil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZIPu_conjunct0) ).

thf(13843,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ ( prod @ A @ B ) ) ) ) @ zip0 ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ nil ) ) )
      = ( s @ ( list @ ( prod @ A @ B ) ) @ nil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1750]) ).

thf(2454,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( prod @ A @ B ) ) @ absu_prod ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ repu_prod ) @ ( s @ ( prod @ A @ B ) @ C ) ) ) ) )
      = ( s @ ( prod @ A @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aprodu_tybiju_conjunct0) ).

thf(17833,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( prod @ A @ B ) ) @ absu_prod ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ repu_prod ) @ ( s @ ( prod @ A @ B ) @ C ) ) ) ) )
      = ( s @ ( prod @ A @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2454]) ).

thf(306,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ toset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
              | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',atoset) ).

thf(5018,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ toset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ poset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
                | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ toset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(3079,axiom,
    ! [A: $i] :
    ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ARCHu_LT) ).

thf(21237,plain,
    ! [A: $i] :
    ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3079]) ).

thf(2780,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NOTu_LE) ).

thf(19584,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2780]) ).

thf(1943,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_ASSOC) ).

thf(15075,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1943]) ).

thf(1564,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_MULT2) ).

thf(12659,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1564]) ).

thf(1519,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ real @ B )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_RNEGu_UNIQ) ).

thf(12364,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ B )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      & ( ( ( s @ real @ B )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1519]) ).

thf(1098,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ C @ B ) @ D ) ) ) ) ) @ ( s @ ( list @ C ) @ F ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( list @ C ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ C @ B ) @ D ) ) ) @ ( s @ ( list @ C ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_o) ).

thf(9567,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( list @ C ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ C @ B ) @ D ) ) ) ) ) @ ( s @ ( list @ C ) @ F ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ C ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( list @ C ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ C @ B ) @ D ) ) ) @ ( s @ ( list @ C ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1098]) ).

thf(2155,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ letu_end ) @ ( s @ A @ B ) ) )
      = ( s @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLETu_ENDu_DEF) ).

thf(16234,plain,
    ! [A: $i,B: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ letu_end ) @ ( s @ A @ B ) ) )
      = ( s @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2155]) ).

thf(1895,axiom,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [C: $i] :
            ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
              & ! [D: $i,E: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ! [D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
             => ( ( s @ ( fun @ A @ bool ) @ C )
                = ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZLu_SUBSETSu_UNIONSu_NONEMPTY) ).

thf(14707,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [C: $i] :
            ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
              & ! [D: $i,E: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ! [D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
             => ( ( s @ ( fun @ A @ bool ) @ C )
                = ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1895]) ).

thf(1502,axiom,
    ! [A: $i] :
      ( ( s @ n10 @ A )
      = ( s @ n10 @ one ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aone) ).

thf(12256,plain,
    ! [A: $i] :
      ( ( s @ n10 @ A )
      = ( s @ n10 @ one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1502]) ).

thf(2964,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_LEu_ADDu_C) ).

thf(20577,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2964]) ).

thf(2157,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_TRIANGLE) ).

thf(16249,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2157]) ).

thf(2883,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_LT) ).

thf(20169,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2883]) ).

thf(2626,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_EMPTY) ).

thf(18771,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2626]) ).

thf(2570,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ A @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDELETEu_DELETE) ).

thf(18439,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ A @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2570]) ).

thf(2847,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ matrix ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_WORKS) ).

thf(19944,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ matrix ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2847]) ).

thf(1312,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ F ) ) ) )
        <=> ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ bool ) ) @ geq ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) )
            <=> ? [I: $i,J: $i,K: $i] :
                  ( ( ( p @ ( s @ bool @ K ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i,I: $i] :
                  ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ A @ I ) ) )
                  = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ A @ I ) ) ) ) ) )
             => ! [G: $i,H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                    & ! [J: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ J ) ) ) ) ) ) ) )
                 => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ I ) ) ) ) )
                    = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ I ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_VSUMu_PRODUCT) ).

thf(10915,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ F ) ) ) )
           => ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ bool ) ) @ geq ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
          & ( ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ bool ) ) @ geq ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ F ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) )
               => ? [I: $i,J: $i,K: $i] :
                    ( ( ( p @ ( s @ bool @ K ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ K ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) )
              & ( ? [I: $i,J: $i,K: $i] :
                    ( ( ( p @ ( s @ bool @ K ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ K ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( prod @ A @ B ) @ H ) ) ) ) ) )
         => ! [F: $i] :
              ( ! [G: $i,H: $i,I: $i] :
                  ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ H ) ) ) @ ( s @ A @ I ) ) )
                  = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ H ) @ ( s @ A @ I ) ) ) ) ) )
             => ! [G: $i,H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                    & ! [J: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) @ ( s @ A @ J ) ) ) ) ) ) ) )
                 => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ I ) ) ) ) )
                    = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ H ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ ( cart @ real @ C ) ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ ( cart @ real @ C ) ) ) @ I ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1312]) ).

thf(2048,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_EQu_TRANS) ).

thf(15645,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2048]) ).

thf(1670,axiom,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) )
    = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_leu_th) ).

thf(13354,plain,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) )
    = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ x1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ y1 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1670]) ).

thf(2775,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_LE) ).

thf(19567,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2775]) ).

thf(1268,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_1u_LT) ).

thf(10636,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1268]) ).

thf(2363,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) )
        = ( s @ real @ B ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_EQ) ).

thf(17249,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) )
          = ( s @ real @ B ) )
       => ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) )
          = ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2363]) ).

thf(817,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_SUC) ).

thf(8060,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[817]) ).

thf(417,axiom,
    ! [A: $i] :
      ( ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ A ) ) )
      = ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anaddu_inv) ).

thf(5767,plain,
    ! [A: $i] :
      ( ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ naddu_inv ) @ ( s @ nadd @ A ) ) )
      = ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).

thf(517,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) )
        = ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPOWERSETu_CLAUSESu_conjunct0) ).

thf(6280,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
     => ( ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) )
        = ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).

thf(1613,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( ( s @ real @ B )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( s @ ( cart @ real @ A ) @ C )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_EQu_0) ).

thf(12937,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ ( cart @ real @ A ) @ C )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
      & ( ( ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ ( cart @ real @ A ) @ C )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1613]) ).

thf(1374,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_SETu_OFu_LIST) ).

thf(11443,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1374]) ).

thf(591,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_TRIANGLEu_HALFu_R) ).

thf(6762,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).

thf(206,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUBu_conjunct9) ).

thf(4346,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(216,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ? [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_DOWN) ).

thf(4398,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ? [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(285,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) ) )
        <=> ? [C: $i,D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ y ) ) ) @ ( s @ nadd @ D ) ) ) ) ) )
     => ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ x ) ) ) @ ( s @ hreal @ y ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ahrealu_mul) ).

thf(4905,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) ) )
           => ? [C: $i,D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ y ) ) ) @ ( s @ nadd @ D ) ) ) ) ) )
          & ( ? [C: $i,D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) ) ) @ ( s @ nadd @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ y ) ) ) @ ( s @ nadd @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ A ) @ ( s @ nadd @ B ) ) ) ) ) )
     => ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ x ) ) ) @ ( s @ hreal @ y ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(2445,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_RADD) ).

thf(17777,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2445]) ).

thf(150,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ B )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_DIVu_RMUL) ).

thf(3940,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ B )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(2347,axiom,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i,E: $i] :
      ( ( s @ ( prod @ A @ B ) @ C )
      = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIRu_SURJECTIVE) ).

thf(17166,plain,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i,E: $i] :
      ( ( s @ ( prod @ A @ B ) @ C )
      = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2347]) ).

thf(688,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_LINV) ).

thf(7264,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[688]) ).

thf(470,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_1) ).

thf(6036,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).

thf(31,axiom,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
      = ( s @ ( list @ A ) @ t0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTL) ).

thf(3305,plain,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
      = ( s @ ( list @ A ) @ t0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(2189,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ m ) ) ) @ ( s @ real @ n ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ n ) ) ) @ ( s @ real @ m ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_ACu_conjunct0) ).

thf(16416,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ m ) ) ) @ ( s @ real @ n ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ n ) ) ) @ ( s @ real @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2189]) ).

thf(3172,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOMPONENT) ).

thf(21708,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3172]) ).

thf(2405,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ B @ F ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONDu_RAND) ).

thf(17558,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ B @ F ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2405]) ).

thf(1822,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_EMPTYu_conjunct0) ).

thf(14319,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1822]) ).

thf(1533,axiom,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ m ) ) ) @ ( s @ hreal @ n ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ n ) ) ) @ ( s @ hreal @ m ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_ADDu_ACu_conjunct0) ).

thf(12455,plain,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ m ) ) ) @ ( s @ hreal @ n ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ n ) ) ) @ ( s @ hreal @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1533]) ).

thf(117,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ m ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ n ) ) ) @ ( s @ real @ p1 ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ n ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ m ) ) ) @ ( s @ real @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_ACu_conjunct2) ).

thf(3791,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ m ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ n ) ) ) @ ( s @ real @ p1 ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ n ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ m ) ) ) @ ( s @ real @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(1106,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ b0 ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ a0 ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ b0 ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_RDISTRIB) ).

thf(9599,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ a0 ) ) ) @ ( s @ real @ b0 ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ a0 ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ b0 ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1106]) ).

thf(2867,axiom,
    ( ( ( s @ bool @ a0 )
      = ( s @ bool @ b0 ) )
  <=> ( ( ( p @ ( s @ bool @ a0 ) )
       => ( p @ ( s @ bool @ b0 ) ) )
      & ( ( p @ ( s @ bool @ b0 ) )
       => ( p @ ( s @ bool @ a0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aGeomdetailo_EQu_EXPAND) ).

thf(20044,plain,
    ( ( ( ( s @ bool @ a0 )
        = ( s @ bool @ b0 ) )
     => ( ( ( p @ ( s @ bool @ a0 ) )
         => ( p @ ( s @ bool @ b0 ) ) )
        & ( ( p @ ( s @ bool @ b0 ) )
         => ( p @ ( s @ bool @ a0 ) ) ) ) )
    & ( ( ( ( p @ ( s @ bool @ a0 ) )
         => ( p @ ( s @ bool @ b0 ) ) )
        & ( ( p @ ( s @ bool @ b0 ) )
         => ( p @ ( s @ bool @ a0 ) ) ) )
     => ( ( s @ bool @ a0 )
        = ( s @ bool @ b0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2867]) ).

thf(617,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ o ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXu_MAP) ).

thf(6912,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ o ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).

thf(1745,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ m ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ n ) ) ) @ ( s @ int @ p1 ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ n ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ m ) ) ) @ ( s @ int @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_ACu_conjunct2) ).

thf(13823,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ m ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ n ) ) ) @ ( s @ int @ p1 ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ n ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ m ) ) ) @ ( s @ int @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1745]) ).

thf(2936,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ A ) ) @ mat ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSPu_MAT) ).

thf(20438,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ A ) ) @ mat ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2936]) ).

thf(3150,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_CROSS) ).

thf(21597,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3150]) ).

thf(2904,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_MIN) ).

thf(20247,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2904]) ).

thf(2015,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_EMPTYu_conjunct0) ).

thf(15519,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2015]) ).

thf(2479,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ A ) )
      | ~ ( p @ ( s @ bool @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXCLUDEDu_MIDDLE) ).

thf(17948,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2479]) ).

thf(1326,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ C )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( s @ real @ A )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_LCANCELu_IMP) ).

thf(11022,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ C )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( s @ real @ A )
        = ( s @ real @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1326]) ).

thf(442,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LETu_ADD) ).

thf(5912,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).

thf(1444,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
          & ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_BOUNDEDu_POS) ).

thf(11915,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
          & ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1444]) ).

thf(912,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ! [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ H ) ) ) ) )
               => ( ( s @ A @ G )
                  = ( s @ A @ H ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_IMAGEu_INJu_GENERAL) ).

thf(8625,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ! [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ H ) ) ) ) )
               => ( ( s @ A @ G )
                  = ( s @ A @ H ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[912]) ).

thf(2416,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ choice0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCHOICE) ).

thf(17586,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ choice0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2416]) ).

thf(1017,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_TRIANGLE) ).

thf(9182,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1017]) ).

thf(1887,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot ) ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ E ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZRECSPACEu_INDUCT) ).

thf(14671,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot ) ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ! [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ num @ F ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ E ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1887]) ).

thf(2037,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_ADDu_LCANCEL) ).

thf(15598,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2037]) ).

thf(2696,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LETu_TOTAL) ).

thf(19106,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2696]) ).

thf(1950,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ f0 ) ) ) @ ( s @ A @ D ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
        = ( s @ ( fun @ A @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEMPTYu_GSPEC) ).

thf(15095,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ f0 ) ) ) @ ( s @ A @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ f0 ) ) ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
        = ( s @ ( fun @ A @ bool ) @ empty ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1950]) ).

thf(2439,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_ASSOC) ).

thf(17745,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2439]) ).

thf(1816,axiom,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INFu_UNIQUE) ).

thf(14279,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) )
        = ( s @ real @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1816]) ).

thf(1282,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_SUBu_REFL) ).

thf(10717,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1282]) ).

thf(2026,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_MUL) ).

thf(15556,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2026]) ).

thf(1242,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_ASu_IMAGEu_SUBSET) ).

thf(10466,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1242]) ).

thf(2284,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAXu_ASSOC) ).

thf(16892,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2284]) ).

thf(1366,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_SUBSET) ).

thf(11374,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1366]) ).

thf(3022,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_IMPu_LE) ).

thf(20883,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3022]) ).

thf(2195,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_CLAUSESu_conjunct4) ).

thf(16461,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2195]) ).

thf(451,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ c ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_LNEG) ).

thf(5948,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ c ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ c ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).

thf(554,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ num @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( cart @ A @ B ) @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_CARTu_UNIV) ).

thf(6513,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ num @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( cart @ A @ B ) @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).

thf(1720,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ n10 ) @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_REAL) ).

thf(13634,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ n10 ) @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1720]) ).

thf(661,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_MULTu_ADD) ).

thf(7149,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[661]) ).

thf(956,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aAPPENDu_conjunct1) ).

thf(8845,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[956]) ).

thf(1062,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUCu_conjunct1) ).

thf(9358,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1062]) ).

thf(1555,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ A ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_FROM) ).

thf(12612,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ A ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1555]) ).

thf(128,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_MONOu_LT) ).

thf(3823,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(1126,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_POS) ).

thf(9815,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1126]) ).

thf(1740,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MINu_MAX) ).

thf(13791,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1740]) ).

thf(307,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_LT) ).

thf(5030,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(1005,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_LEu_1) ).

thf(9082,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1005]) ).

thf(2246,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ A ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJOINTu_NUMSEG) ).

thf(16695,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2246]) ).

thf(113,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
         => ( ( ( s @ A @ C )
              = ( s @ A @ D ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_TOTALu_LT) ).

thf(3774,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) )
         => ( ( ( s @ A @ C )
              = ( s @ A @ D ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(2046,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_TRIANGLEu_LE) ).

thf(15640,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2046]) ).

thf(3161,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_ASSOC) ).

thf(21650,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3161]) ).

thf(863,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_MULTu_conjunct0) ).

thf(8327,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[863]) ).

thf(954,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_ADDBOUND) ).

thf(8840,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[954]) ).

thf(830,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) @ ( s @ ( fun @ num @ num ) @ suc ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_SUC) ).

thf(8148,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ o ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ B ) ) ) @ ( s @ ( fun @ num @ num ) @ suc ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[830]) ).

thf(130,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SGNu_0) ).

thf(3828,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(1335,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_MONOu_LT) ).

thf(11109,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1335]) ).

thf(2310,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ union ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ union ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPPORTu_CLAUSESu_conjunct3) ).

thf(16987,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ union ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ union ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2310]) ).

thf(2108,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMu_FILTER) ).

thf(15969,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2108]) ).

thf(1048,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_LEu_MUL) ).

thf(9313,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1048]) ).

thf(847,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ C )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( ( s @ real @ A )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_EQu_RCANCELu_IMP) ).

thf(8248,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ C )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( ( s @ real @ A )
        = ( s @ real @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[847]) ).

thf(2598,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_ADD) ).

thf(18625,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) ) )
      & ( ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2598]) ).

thf(1075,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',avectoru_neg) ).

thf(9413,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1075]) ).

thf(1342,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_POWERSET) ).

thf(11198,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
               => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
              & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1342]) ).

thf(1330,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_RSQRT) ).

thf(11034,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1330]) ).

thf(294,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_MULTu_conjunct5) ).

thf(4982,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(2950,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_MULu_FINITE) ).

thf(20506,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2950]) ).

thf(2559,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_TRANS) ).

thf(18387,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2559]) ).

thf(1996,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_ADDu_LCANCEL) ).

thf(15411,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1996]) ).

thf(2081,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) )
     => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_LINEAR) ).

thf(15785,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) )
     => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2081]) ).

thf(1158,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ B @ G ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aiterate) ).

thf(9963,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ B @ G ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
         => ! [E: $i,F: $i,G: $i] :
              ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) ) @ D ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1158]) ).

thf(2437,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
        & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_ANDu_FORALLu_THM) ).

thf(17730,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( p @ ( s @ bool @ B ) )
          & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ( p @ ( s @ bool @ B ) )
          & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2437]) ).

thf(1548,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i] :
        ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LINVu_LEMMA2) ).

thf(12580,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i] :
        ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1548]) ).

thf(2097,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_HULL) ).

thf(15888,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2097]) ).

thf(2057,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_LT) ).

thf(15676,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2057]) ).

thf(319,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_0) ).

thf(5106,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).

thf(3083,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_TRANS) ).

thf(21266,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3083]) ).

thf(111,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EXPu_conjunct4) ).

thf(3769,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(1948,axiom,
    ! [A: $i] :
      ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ minimal ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) ) ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ minimal ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) ) ) )
           => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMINIMAL) ).

thf(15085,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ minimal ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) ) ) )
          & ! [B: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ minimal ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) ) ) )
             => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ minimal ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) ) ) )
          & ! [B: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ minimal ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) ) ) )
             => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ A ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1948]) ).

thf(2788,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_LNEG) ).

thf(19613,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2788]) ).

thf(2807,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) )
     => ( ( s @ hreal @ B )
        = ( s @ hreal @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_ADDu_LCANCEL) ).

thf(19712,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) )
        = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) )
     => ( ( s @ hreal @ B )
        = ( s @ hreal @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2807]) ).

thf(1086,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) )
     => ( ! [C: $i] :
          ? [D: $i] :
            ( ( ( p @ ( s @ bool @ D ) )
            <=> ( ( s @ num @ C )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ f ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ f ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ f ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ f ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_CLAUSESu_NUMSEG) ).

thf(9505,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) )
     => ( ! [C: $i] :
          ? [D: $i] :
            ( ( ( p @ ( s @ bool @ D ) )
             => ( ( s @ num @ C )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
            & ( ( ( s @ num @ C )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
             => ( p @ ( s @ bool @ D ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ f ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ f ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ f ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ A ) @ A ) ) ) @ iterate ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ f ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1086]) ).

thf(1381,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) )
    <=> ? [D: $i,E: $i] :
          ( ( ( s @ ( list @ A ) @ B )
            = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
          & ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ E ) ) )
            = ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLENGTHu_EQu_CONS) ).

thf(11486,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) )
       => ? [D: $i,E: $i] :
            ( ( ( s @ ( list @ A ) @ B )
              = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
            & ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ E ) ) )
              = ( s @ num @ C ) ) ) )
      & ( ? [D: $i,E: $i] :
            ( ( ( s @ ( list @ A ) @ B )
              = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
            & ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ E ) ) )
              = ( s @ num @ C ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1381]) ).

thf(2557,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_FINITE) ).

thf(18377,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2557]) ).

thf(233,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_conjunct1) ).

thf(4510,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(1207,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) ) ) )
        <=> ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) )
              = ( s @ A @ D ) )
            & ! [G: $i,H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ A ) ) ) @ itset ) @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ A ) ) @ h_ ) @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITSET) ).

thf(10231,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) ) ) )
           => ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) )
                = ( s @ A @ D ) )
              & ! [G: $i,H: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                 => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) ) ) ) ) ) )
          & ( ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ empty ) ) )
                = ( s @ A @ D ) )
              & ! [G: $i,H: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                 => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ insert ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ E ) @ ( s @ B @ G ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ A ) ) ) @ itset ) @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ A ) ) @ h_ ) @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ B @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ A ) ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1207]) ).

thf(1872,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) )
     => ! [E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
          ( ( ! [K: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ K ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ J ) @ ( s @ B @ K ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ J ) @ ( s @ B @ K ) ) ) ) )
                    = ( s @ B @ K ) ) ) )
            & ! [K: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ K ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ J ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ K ) ) ) ) )
                    = ( s @ A @ K ) )
                  & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ H ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ K ) ) ) ) )
                    = ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ K ) ) ) ) ) ) )
         => ( ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ C ) @ G ) ) )
            = ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_EQu_GENERALu_INVERSES) ).

thf(14607,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) )
     => ! [E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
          ( ( ! [K: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ K ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ J ) @ ( s @ B @ K ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ J ) @ ( s @ B @ K ) ) ) ) )
                    = ( s @ B @ K ) ) ) )
            & ! [K: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ K ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ J ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ K ) ) ) ) )
                    = ( s @ A @ K ) )
                  & ( ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ H ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ I ) @ ( s @ A @ K ) ) ) ) )
                    = ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ K ) ) ) ) ) ) )
         => ( ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ C ) @ G ) ) )
            = ( s @ C @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ C ) @ C ) ) ) @ iterate ) @ ( s @ ( fun @ C @ ( fun @ C @ C ) ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1872]) ).

thf(926,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
        = ( s @ real @ A ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_REFL) ).

thf(8679,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
          = ( s @ real @ A ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
          = ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[926]) ).

thf(2285,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSEGu_SUBSETu_FL) ).

thf(16895,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2285]) ).

thf(2870,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ c0 ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ c0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_ACu_conjunct1) ).

thf(20106,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ c0 ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ c0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2870]) ).

thf(1021,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LTu_1) ).

thf(9194,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1021]) ).

thf(1144,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_LID) ).

thf(9891,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1144]) ).

thf(1947,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ B ) )
        | ! [D: $i] : ( p @ ( s @ bool @ C ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
          | ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRIVu_ORu_FORALLu_THM) ).

thf(15084,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1947]) ).

thf(2618,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) )
         => ( ( s @ B @ C )
            = ( s @ B @ D ) ) )
    <=> ? [C: $i] :
        ! [D: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
          = ( s @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJECTIVEu_LEFTu_INVERSE) ).

thf(18720,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i,D: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) )
           => ( ( s @ B @ C )
              = ( s @ B @ D ) ) )
       => ? [C: $i] :
          ! [D: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
            = ( s @ B @ D ) ) )
      & ( ? [C: $i] :
          ! [D: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
            = ( s @ B @ D ) )
       => ! [C: $i,D: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) )
           => ( ( s @ B @ C )
              = ( s @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2618]) ).

thf(1525,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) )
        <=> ( ( ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) )
              = ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
            & ( ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ matrixu_inv ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
          = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ h_ ) @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amatrixu_inv) ).

thf(12416,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) )
           => ( ( ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) )
                = ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
              & ( ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
                = ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
          & ( ( ( ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) )
                = ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
              & ( ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ A ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
                = ( s @ ( cart @ ( cart @ real @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ A ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ matrixu_inv ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
          = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ h_ ) @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ bool ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1525]) ).

thf(2545,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ t ) )
    <=> ( ( s @ ( fun @ ( finite_image @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( finite_image @ A ) @ B ) ) @ destu_cart ) @ ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ A ) @ B ) @ ( cart @ B @ A ) ) @ mku_cart ) @ ( s @ ( fun @ ( finite_image @ A ) @ B ) @ C ) ) ) ) )
        = ( s @ ( fun @ ( finite_image @ A ) @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',acartu_tybiju_conjunct1) ).

thf(18330,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ t ) )
       => ( ( s @ ( fun @ ( finite_image @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( finite_image @ A ) @ B ) ) @ destu_cart ) @ ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ A ) @ B ) @ ( cart @ B @ A ) ) @ mku_cart ) @ ( s @ ( fun @ ( finite_image @ A ) @ B ) @ C ) ) ) ) )
          = ( s @ ( fun @ ( finite_image @ A ) @ B ) @ C ) ) )
      & ( ( ( s @ ( fun @ ( finite_image @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ B @ A ) @ ( fun @ ( finite_image @ A ) @ B ) ) @ destu_cart ) @ ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ A ) @ B ) @ ( cart @ B @ A ) ) @ mku_cart ) @ ( s @ ( fun @ ( finite_image @ A ) @ B ) @ C ) ) ) ) )
          = ( s @ ( fun @ ( finite_image @ A ) @ B ) @ C ) )
       => ( p @ ( s @ bool @ t ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2545]) ).

thf(1340,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
                = ( s @ A @ G ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) )
                = ( s @ B @ G ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIJECTIONSu_CARDu_EQ) ).

thf(11181,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
                = ( s @ A @ G ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) )
                = ( s @ B @ G ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1340]) ).

thf(1585,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPOWu_2u_SQRTu_ABS) ).

thf(12754,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1585]) ).

thf(1084,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_CLOSED) ).

thf(9495,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1084]) ).

thf(705,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
    <=> ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
        & ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_MULu_1) ).

thf(7348,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
       => ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
      & ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
          & ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).

thf(1255,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ y ) ) ) @ ( s @ real @ z ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ y ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ z ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_ACIu_conjunct2) ).

thf(10536,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ y ) ) ) @ ( s @ real @ z ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ y ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ z ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1255]) ).

thf(2885,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_tybiju_conjunct0) ).

thf(20178,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2885]) ).

thf(2333,axiom,
    ! [A: $i,B: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPSUBSETu_IRREFL) ).

thf(17099,plain,
    ! [A: $i,B: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2333]) ).

thf(159,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_1u_LT) ).

thf(3970,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(41,axiom,
    ! [A: $i] :
      ( ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_NEGTOTAL) ).

thf(3415,plain,
    ! [A: $i] :
      ( ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(2795,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
    <=> ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_SUBSETu_IMAGE) ).

thf(19650,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
       => ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) )
      & ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) )
       => ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2795]) ).

thf(2112,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_ACu_conjunct1) ).

thf(15982,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ x ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2112]) ).

thf(2377,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSEGu_WOSET) ).

thf(17308,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2377]) ).

thf(183,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ num ) @ E ) ) ) )
        <=> ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ geq ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ num ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) )
            <=> ? [H: $i,I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ J ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                 => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ H ) ) ) ) )
                    = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ num ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ H ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_NSUMu_PRODUCT) ).

thf(4133,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ num ) @ E ) ) ) )
           => ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ geq ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ num ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
          & ( ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ geq ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ num ) @ E ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ num ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) )
               => ? [H: $i,I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) )
              & ( ? [H: $i,I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( prod @ A @ B ) @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ! [I: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                 => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ H ) ) ) ) )
                    = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ num ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ num ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ H ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(1684,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asetu_ofu_listu_conjunct1) ).

thf(13446,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1684]) ).

thf(2544,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_UNIONu_LEFT) ).

thf(18327,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2544]) ).

thf(1216,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_CONVEXu_BOUNDu_LE) ).

thf(10293,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1216]) ).

thf(2088,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
        <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_EX) ).

thf(15837,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) )
          & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( list @ B ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2088]) ).

thf(534,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ g ) @ ( s @ A @ E ) ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ g ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_EQu_SUPERSET) ).

thf(6369,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ g ) @ ( s @ A @ E ) ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ g ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).

thf(1060,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LEu_conjunct2) ).

thf(9353,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1060]) ).

thf(559,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ A ) )
    <=> ( ( s @ num @ B )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_ADDu_LCANCELu_0) ).

thf(6526,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ A ) )
       => ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).

thf(2354,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_ACIu_conjunct4) ).

thf(17188,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2354]) ).

thf(1064,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) ) @ row ) @ ( s @ num @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aROWu_TRANSP) ).

thf(9363,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ real @ B ) ) ) @ row ) @ ( s @ num @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1064]) ).

thf(2707,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
       => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_IMPu_EXISTSu_THM) ).

thf(19169,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( p @ ( s @ bool @ B ) )
         => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ B ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ( p @ ( s @ bool @ B ) )
         => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2707]) ).

thf(297,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_LDISTRIB) ).

thf(4991,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(498,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ s0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_1) ).

thf(6193,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ s0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).

thf(2761,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ empty ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFu_SING) ).

thf(19435,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ empty ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2761]) ).

thf(2577,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D )
        = ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) )
    <=> ( ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ B ) ) @ fstcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ) )
          = ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ B ) ) @ fstcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) )
        & ( ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ C ) ) @ sndcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ) )
          = ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ C ) ) @ sndcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPASTECARTu_EQ) ).

thf(18478,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D )
          = ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) )
       => ( ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ B ) ) @ fstcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ) )
            = ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ B ) ) @ fstcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) )
          & ( ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ C ) ) @ sndcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ) )
            = ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ C ) ) @ sndcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) ) ) )
      & ( ( ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ B ) ) @ fstcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ) )
            = ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ B ) ) @ fstcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) )
          & ( ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ C ) ) @ sndcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ) )
            = ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ C ) ) @ sndcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) ) )
       => ( ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D )
          = ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2577]) ).

thf(2827,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_TRANS) ).

thf(19844,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2827]) ).

thf(3105,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_LEu_REFL) ).

thf(21397,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3105]) ).

thf(2836,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_NEG) ).

thf(19888,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2836]) ).

thf(1878,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) ) @ D ) @ ( s @ ( fun @ B @ real ) @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) @ ( s @ B @ H ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ E ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ F ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) @ ( s @ B @ H ) ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) @ ( s @ B @ H ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ F ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) @ ( s @ B @ H ) ) ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) )
             => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ F ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ B @ real ) @ H ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ I ) ) ) ) ) ) )
                = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) ) @ D ) @ ( s @ ( fun @ B @ real ) @ H ) ) ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ I ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_VSUMu_MUL) ).

thf(14631,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) ) @ D ) @ ( s @ ( fun @ B @ real ) @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) @ ( s @ B @ H ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ E ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ F ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) @ ( s @ B @ H ) ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ B @ real ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) @ ( s @ B @ H ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ F ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) @ ( s @ B @ H ) ) ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) ) )
             => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ F ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ B @ real ) @ H ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ I ) ) ) ) ) ) )
                = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ ( fun @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( fun @ B @ ( cart @ real @ A ) ) ) ) ) @ D ) @ ( s @ ( fun @ B @ real ) @ H ) ) ) @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ I ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1878]) ).

thf(2257,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        | ( ( s @ num @ A )
          = ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_LT) ).

thf(16763,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          | ( ( s @ num @ A )
            = ( s @ num @ B ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          | ( ( s @ num @ A )
            = ( s @ num @ B ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2257]) ).

thf(2814,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_FLEQ) ).

thf(19742,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2814]) ).

thf(1756,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_SUBSET) ).

thf(13870,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1756]) ).

thf(1852,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_LEu_LADDu_IMP) ).

thf(14507,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1852]) ).

thf(1032,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_abs) ).

thf(9242,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1032]) ).

thf(2970,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ union ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERSu_UNION) ).

thf(20599,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ union ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2970]) ).

thf(2610,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_UNIONu_EQ) ).

thf(18692,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2610]) ).

thf(1278,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ! [J: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
                  & ! [J: $i] :
                      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                     => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) )
                        = ( s @ B @ F ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ bool @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( fun @ A @ B ) @ H ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ num @ G ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ num @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ B @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ E ) ) ) @ ( s @ num @ G ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_FUNSPACE) ).

thf(10663,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ! [J: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
                    & ! [J: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                       => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) )
                          = ( s @ B @ F ) ) ) ) )
                & ( ( ! [J: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
                    & ! [J: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                       => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) )
                          = ( s @ B @ F ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ bool @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( fun @ A @ B ) @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ! [J: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
                    & ! [J: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                       => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) )
                          = ( s @ B @ F ) ) ) ) )
                & ( ( ! [J: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
                    & ! [J: $i] :
                        ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                       => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) )
                          = ( s @ B @ F ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ bool @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ ( fun @ A @ B ) @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ num @ G ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ num @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) @ ( s @ B @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ E ) ) ) @ ( s @ num @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1278]) ).

thf(3101,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_EQu_SYM) ).

thf(21379,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3101]) ).

thf(1163,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_MULT) ).

thf(9983,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1163]) ).

thf(2650,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ null ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNULLu_conjunct0) ).

thf(18858,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ null ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2650]) ).

thf(1886,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) )
     => ? [B: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_FINITEu_AVOID) ).

thf(14668,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) )
     => ? [B: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1886]) ).

thf(336,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUBu_conjunct8) ).

thf(5226,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(480,axiom,
    ! [A: $i] :
      ( ? [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ C ) ) ) )
          & ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) )
    <=> ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_ODDu_DECOMPOSITION) ).

thf(6086,plain,
    ! [A: $i] :
      ( ( ? [B: $i,C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ C ) ) ) )
            & ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) )
       => ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ? [B: $i,C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ C ) ) ) )
            & ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).

thf(257,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_gt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_ge ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_GTu_DISCRETE) ).

thf(4695,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_gt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_ge ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(1362,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) )
    <=> ( ( s @ num @ ( i @ ( s @ ( fun @ ( finite_image @ A ) @ num ) @ destu_finiteu_image ) @ ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',afiniteu_imageu_tybiju_conjunct1) ).

thf(11349,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ ( finite_image @ A ) @ num ) @ destu_finiteu_image ) @ ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ B ) ) ) ) )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( finite_image @ A ) @ num ) @ destu_finiteu_image ) @ ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ B ) ) ) ) )
          = ( s @ num @ B ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ bool ) ) @ in ) @ ( s @ num @ B ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1362]) ).

thf(988,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ C ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_DELETEu_CASES) ).

thf(9019,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ C ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[988]) ).

thf(2160,axiom,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTEu_ANTISYM) ).

thf(16262,plain,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2160]) ).

thf(1677,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) )
        <=> ? [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_ELIMu_PAIRu_THM) ).

thf(13374,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) )
           => ? [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
          & ( ? [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ bool @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ setspec ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( prod @ A @ B ) @ E ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1677]) ).

thf(2323,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_MIN) ).

thf(17044,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2323]) ).

thf(151,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_DIV) ).

thf(3943,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(2823,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_RDISTRIB) ).

thf(19791,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2823]) ).

thf(1954,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
         => ? [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) )
                = ( s @ ( fun @ B @ bool ) @ F ) ) ) )
    <=> ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
         => ? [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ G ) ) )
                = ( s @ B @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJECTIVEu_ONu_IMAGE) ).

thf(15116,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) )
                  = ( s @ ( fun @ B @ bool ) @ F ) ) ) )
       => ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ G ) ) )
                  = ( s @ B @ F ) ) ) ) )
      & ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ G ) ) )
                  = ( s @ B @ F ) ) ) )
       => ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) )
                  = ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1954]) ).

thf(1689,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GEu_conjunct4) ).

thf(13460,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1689]) ).

thf(623,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        | ( ( s @ num @ C )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_MULTu_RCANCEL) ).

thf(6943,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          | ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          | ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).

thf(814,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
        = ( s @ int @ A ) )
    <=> ( ( s @ int @ B )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_EQu_ADDu_LCANCELu_0) ).

thf(8043,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ A ) )
       => ( ( s @ int @ B )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ int @ B )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[814]) ).

thf(370,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_SUC) ).

thf(5472,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).

thf(543,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i] :
        ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LINVu_LEMMA4) ).

thf(6467,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i] :
        ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).

thf(1748,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_LINV) ).

thf(13831,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1748]) ).

thf(824,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_BOUND) ).

thf(8087,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[824]) ).

thf(3176,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_ABS) ).

thf(21726,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3176]) ).

thf(19,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h2 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ h2 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALL2u_conjunct3) ).

thf(3256,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h2 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ h2 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ h2 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( list @ B ) @ ( list @ B ) ) ) @ cons ) @ ( s @ B @ h2 ) ) ) @ ( s @ ( list @ B ) @ t2 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(1475,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ real @ A )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_EQu_0) ).

thf(12054,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ real @ A )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1475]) ).

thf(1925,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
         => ( ( s @ B @ E )
            = ( s @ B @ F ) ) )
    <=> ? [E: $i] :
        ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
            = ( s @ B @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJECTIVEu_ONu_LEFTu_INVERSE) ).

thf(14992,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
           => ( ( s @ B @ E )
              = ( s @ B @ F ) ) )
       => ? [E: $i] :
          ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
              = ( s @ B @ F ) ) ) )
      & ( ? [E: $i] :
          ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
              = ( s @ B @ F ) ) )
       => ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ F ) ) ) ) )
           => ( ( s @ B @ E )
              = ( s @ B @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1925]) ).

thf(3164,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C )
      = ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIRu_EXISTSu_THM) ).

thf(21661,plain,
    ! [A: $i,B: $i] :
    ? [C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C )
      = ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ mku_pair ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3164]) ).

thf(509,axiom,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y1 ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y2 ) ) ) ) )
   => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y2 ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y2 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y2 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRATu_LEMMA3) ).

thf(6245,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y1 ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ y2 ) ) ) ) )
   => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y2 ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y2 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x2 ) ) ) @ ( s @ real @ y1 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y1 ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ y2 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).

thf(2912,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_SUBu_RADD) ).

thf(20279,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2912]) ).

thf(2920,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ A )
        = ( s @ real @ B ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_TOTAL) ).

thf(20343,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ A )
        = ( s @ real @ B ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2920]) ).

thf(1067,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ A @ A ) @ i1 ) ) )
      = ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ i1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_I) ).

thf(9374,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ A @ A ) @ i1 ) ) )
      = ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ i1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1067]) ).

thf(272,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ numu_divides ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_divides) ).

thf(4795,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ numu_divides ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(831,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_SQUAREu_ABS) ).

thf(8151,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[831]) ).

thf(2067,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONDu_ABS) ).

thf(15717,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2067]) ).

thf(910,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i] :
          ( ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ E @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ I ) ) ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) )
          = ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ F ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ F @ B ) @ ( fun @ ( list @ F ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ F @ B ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ H ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) ) ) ) @ ( s @ ( list @ F ) @ ( i @ ( s @ ( fun @ E @ ( list @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i] :
              ( ( s @ ( fun @ E @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ J ) ) ) @ ( s @ ( fun @ C @ D ) @ K ) ) )
              = ( s @ ( fun @ E @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ J ) ) ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ J ) ) ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ F @ E ) @ B ) @ L ) ) ) )
                <=> ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ F @ E ) @ B ) @ L ) @ ( s @ ( prod @ F @ E ) @ ( i @ ( s @ ( fun @ E @ ( prod @ F @ E ) ) @ ( i @ ( s @ ( fun @ F @ ( fun @ E @ ( prod @ F @ E ) ) ) @ c_ ) @ ( s @ F @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ F @ B ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ E @ N ) ) ) @ ( s @ F @ M ) ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i] :
                      ( ( s @ ( fun @ ( prod @ F @ E ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ K ) ) ) @ ( s @ ( fun @ C @ D ) @ L ) ) )
                      = ( s @ ( fun @ ( prod @ F @ E ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ K ) ) ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) ) ) )
                 => ! [K: $i] :
                      ( ! [L: $i,M: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ L ) ) ) @ ( s @ ( fun @ ( prod @ F @ E ) @ A ) @ M ) ) ) )
                        <=> ! [N: $i,O: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ F @ E ) @ A ) @ M ) @ ( s @ ( prod @ F @ E ) @ ( i @ ( s @ ( fun @ E @ ( prod @ F @ E ) ) @ ( i @ ( s @ ( fun @ F @ ( fun @ E @ ( prod @ F @ E ) ) ) @ c_ ) @ ( s @ F @ N ) ) ) @ ( s @ E @ O ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ L ) @ ( s @ E @ O ) ) ) ) ) ) )
                     => ! [L: $i] :
                          ( ! [M: $i,N: $i,O: $i,P: $i] :
                              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ N ) ) ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ ( fun @ ( prod @ F @ E ) @ bool ) @ P ) ) ) )
                            <=> ! [Q: $i,R: $i] :
                                ? [S: $i] :
                                  ( ( ( p @ ( s @ bool @ S ) )
                                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ E @ R ) ) ) )
                                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ F ) @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ ( list @ F ) @ bool ) ) @ mem ) @ ( s @ F @ Q ) ) ) @ ( s @ ( list @ F ) @ ( i @ ( s @ ( fun @ E @ ( list @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ N ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ E @ R ) ) ) ) ) ) ) )
                                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ F @ E ) @ bool ) @ P ) @ ( s @ ( prod @ F @ E ) @ ( i @ ( s @ ( fun @ E @ ( prod @ F @ E ) ) @ ( i @ ( s @ ( fun @ F @ ( fun @ E @ ( prod @ F @ E ) ) ) @ c_ ) @ ( s @ F @ Q ) ) ) @ ( s @ E @ R ) ) ) ) ) ) ) @ ( s @ bool @ S ) ) ) ) ) )
                         => ! [M: $i] :
                              ( ! [N: $i,O: $i,P: $i] :
                                  ( ( s @ ( fun @ ( prod @ F @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ O ) ) ) @ ( s @ ( fun @ C @ D ) @ P ) ) )
                                  = ( s @ ( fun @ ( prod @ F @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ O ) ) ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) ) ) )
                             => ! [N: $i,O: $i,P: $i,Q: $i,R: $i] :
                                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ E @ A ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ R ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ R ) ) ) ) ) @ ( s @ ( fun @ ( prod @ F @ E ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) @ ( fun @ ( prod @ F @ E ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ P ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ Q ) ) ) ) ) ) )
                                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ E @ A ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_MAP) ).

thf(8565,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i] :
          ( ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ E @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ I ) ) ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) )
          = ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ F ) @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ F @ B ) @ ( fun @ ( list @ F ) @ ( list @ B ) ) ) @ map0 ) @ ( s @ ( fun @ F @ B ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ H ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) ) ) ) @ ( s @ ( list @ F ) @ ( i @ ( s @ ( fun @ E @ ( list @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ J ) ) ) @ ( s @ E @ K ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i] :
              ( ( s @ ( fun @ E @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ J ) ) ) @ ( s @ ( fun @ C @ D ) @ K ) ) )
              = ( s @ ( fun @ E @ ( list @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ J ) ) ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ J ) ) ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ F @ E ) @ B ) @ L ) ) ) )
                   => ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ F @ E ) @ B ) @ L ) @ ( s @ ( prod @ F @ E ) @ ( i @ ( s @ ( fun @ E @ ( prod @ F @ E ) ) @ ( i @ ( s @ ( fun @ F @ ( fun @ E @ ( prod @ F @ E ) ) ) @ c_ ) @ ( s @ F @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ F @ B ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ E @ N ) ) ) @ ( s @ F @ M ) ) ) ) ) ) )
                  & ( ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ F @ E ) @ B ) @ L ) @ ( s @ ( prod @ F @ E ) @ ( i @ ( s @ ( fun @ E @ ( prod @ F @ E ) ) @ ( i @ ( s @ ( fun @ F @ ( fun @ E @ ( prod @ F @ E ) ) ) @ c_ ) @ ( s @ F @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ F @ B ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ E @ N ) ) ) @ ( s @ F @ M ) ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ J ) ) ) @ ( s @ ( fun @ C @ D ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ F @ E ) @ B ) @ L ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i] :
                      ( ( s @ ( fun @ ( prod @ F @ E ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ K ) ) ) @ ( s @ ( fun @ C @ D ) @ L ) ) )
                      = ( s @ ( fun @ ( prod @ F @ E ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ B ) @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ K ) ) ) @ ( s @ ( fun @ C @ D ) @ L ) ) ) ) ) )
                 => ! [K: $i] :
                      ( ! [L: $i,M: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ L ) ) ) @ ( s @ ( fun @ ( prod @ F @ E ) @ A ) @ M ) ) ) )
                           => ! [N: $i,O: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ F @ E ) @ A ) @ M ) @ ( s @ ( prod @ F @ E ) @ ( i @ ( s @ ( fun @ E @ ( prod @ F @ E ) ) @ ( i @ ( s @ ( fun @ F @ ( fun @ E @ ( prod @ F @ E ) ) ) @ c_ ) @ ( s @ F @ N ) ) ) @ ( s @ E @ O ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ L ) @ ( s @ E @ O ) ) ) ) ) ) )
                          & ( ! [N: $i,O: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ F @ E ) @ A ) @ M ) @ ( s @ ( prod @ F @ E ) @ ( i @ ( s @ ( fun @ E @ ( prod @ F @ E ) ) @ ( i @ ( s @ ( fun @ F @ ( fun @ E @ ( prod @ F @ E ) ) ) @ c_ ) @ ( s @ F @ N ) ) ) @ ( s @ E @ O ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ L ) @ ( s @ E @ O ) ) ) ) ) )
                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ L ) ) ) @ ( s @ ( fun @ ( prod @ F @ E ) @ A ) @ M ) ) ) ) ) )
                     => ! [L: $i] :
                          ( ! [M: $i,N: $i,O: $i,P: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ N ) ) ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ ( fun @ ( prod @ F @ E ) @ bool ) @ P ) ) ) )
                               => ! [Q: $i,R: $i] :
                                  ? [S: $i] :
                                    ( ( ( p @ ( s @ bool @ S ) )
                                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ E @ R ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ F ) @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ ( list @ F ) @ bool ) ) @ mem ) @ ( s @ F @ Q ) ) ) @ ( s @ ( list @ F ) @ ( i @ ( s @ ( fun @ E @ ( list @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ N ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ E @ R ) ) ) ) ) ) ) )
                                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ E @ R ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ F ) @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ ( list @ F ) @ bool ) ) @ mem ) @ ( s @ F @ Q ) ) ) @ ( s @ ( list @ F ) @ ( i @ ( s @ ( fun @ E @ ( list @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ N ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ E @ R ) ) ) ) ) ) )
                                     => ( p @ ( s @ bool @ S ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ F @ E ) @ bool ) @ P ) @ ( s @ ( prod @ F @ E ) @ ( i @ ( s @ ( fun @ E @ ( prod @ F @ E ) ) @ ( i @ ( s @ ( fun @ F @ ( fun @ E @ ( prod @ F @ E ) ) ) @ c_ ) @ ( s @ F @ Q ) ) ) @ ( s @ E @ R ) ) ) ) ) ) ) @ ( s @ bool @ S ) ) ) ) ) )
                              & ( ! [Q: $i,R: $i] :
                                  ? [S: $i] :
                                    ( ( ( p @ ( s @ bool @ S ) )
                                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ E @ R ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ F ) @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ ( list @ F ) @ bool ) ) @ mem ) @ ( s @ F @ Q ) ) ) @ ( s @ ( list @ F ) @ ( i @ ( s @ ( fun @ E @ ( list @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ N ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ E @ R ) ) ) ) ) ) ) )
                                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ E @ R ) ) ) )
                                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ F ) @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ ( list @ F ) @ bool ) ) @ mem ) @ ( s @ F @ Q ) ) ) @ ( s @ ( list @ F ) @ ( i @ ( s @ ( fun @ E @ ( list @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ N ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ E @ R ) ) ) ) ) ) )
                                     => ( p @ ( s @ bool @ S ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ F @ E ) @ bool ) @ P ) @ ( s @ ( prod @ F @ E ) @ ( i @ ( s @ ( fun @ E @ ( prod @ F @ E ) ) @ ( i @ ( s @ ( fun @ F @ ( fun @ E @ ( prod @ F @ E ) ) ) @ c_ ) @ ( s @ F @ Q ) ) ) @ ( s @ E @ R ) ) ) ) ) ) ) @ ( s @ bool @ S ) ) ) ) )
                               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ N ) ) ) @ ( s @ ( fun @ C @ D ) @ O ) ) ) @ ( s @ ( fun @ ( prod @ F @ E ) @ bool ) @ P ) ) ) ) ) )
                         => ! [M: $i] :
                              ( ! [N: $i,O: $i,P: $i] :
                                  ( ( s @ ( fun @ ( prod @ F @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ O ) ) ) @ ( s @ ( fun @ C @ D ) @ P ) ) )
                                  = ( s @ ( fun @ ( prod @ F @ E ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ bool ) @ bool ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ O ) ) ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) ) ) )
                             => ! [N: $i,O: $i,P: $i,Q: $i,R: $i] :
                                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ E @ A ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ R ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ R ) ) ) ) ) @ ( s @ ( fun @ ( prod @ F @ E ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) @ ( fun @ ( prod @ F @ E ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( prod @ F @ E ) @ A ) @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ P ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ ( prod @ F @ E ) @ B ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ Q ) ) ) ) ) ) )
                                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ E @ A ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ B ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( fun @ F @ B ) ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ ( list @ F ) ) ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[910]) ).

thf(2488,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
     != ( s @ ( fun @ A @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_INSERTu_EMPTY) ).

thf(17987,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
     != ( s @ ( fun @ A @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2488]) ).

thf(1198,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_BOUNDu_COMPONENTu_LT) ).

thf(10178,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1198]) ).

thf(1854,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBETAu_THM) ).

thf(14518,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1854]) ).

thf(235,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
          & ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ D ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQUAREu_CONTINUOUS) ).

thf(4516,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
          & ! [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ D ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(2417,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_TRIANGLE) ).

thf(17589,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2417]) ).

thf(2613,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ) )
     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_REALu_IMPu_UNCOUNTABLE) ).

thf(18707,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ) )
     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2613]) ).

thf(1148,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( ( s @ real @ B )
        = ( s @ real @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_EQ) ).

thf(9902,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( ( s @ real @ B )
        = ( s @ real @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1148]) ).

thf(25,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ min ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_MIN) ).

thf(3288,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ min ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(537,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_DIFF) ).

thf(6420,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).

thf(1980,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_MULT2) ).

thf(15283,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1980]) ).

thf(1446,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ D )
         != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
     => ( ( s @ real @ B )
        = ( s @ real @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_RCANCELu_IMP) ).

thf(11921,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ D )
         != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
     => ( ( s @ real @ B )
        = ( s @ real @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1446]) ).

thf(504,axiom,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GTu_conjunct3) ).

thf(6220,plain,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).

thf(214,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ D ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ f ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ f ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ f ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_UNIONu_EQ) ).

thf(4392,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
        & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ D ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ f ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ f ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ f ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(2548,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_MULu_WELLDEF) ).

thf(18345,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2548]) ).

thf(709,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) ) ) )
           => ( ( s @ A @ E )
              = ( s @ A @ F ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ num ) ) ) @ o ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_INJECTION) ).

thf(7380,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ C ) @ ( s @ A @ F ) ) ) ) )
           => ( ( s @ A @ E )
              = ( s @ A @ F ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ A ) @ ( fun @ A @ num ) ) ) @ o ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[709]) ).

thf(1281,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_max) ).

thf(10714,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1281]) ).

thf(2527,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_LEu_TOTAL) ).

thf(18211,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2527]) ).

thf(444,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_SUPPORT) ).

thf(5917,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).

thf(1579,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_ONE) ).

thf(12722,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1579]) ).

thf(2004,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_NEG2) ).

thf(15469,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2004]) ).

thf(2730,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_MUL) ).

thf(19285,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2730]) ).

thf(1204,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_NZ) ).

thf(10214,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ A )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) )
       => ( ( s @ real @ A )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1204]) ).

thf(867,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i,I: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) @ ( s @ D @ H ) ) ) @ ( s @ C @ I ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ H ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i] :
              ( ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ H ) ) ) @ ( s @ D @ I ) ) )
              = ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ H ) ) ) @ ( s @ D @ I ) ) ) )
         => ! [G: $i] :
              ( ! [H: $i,I: $i] :
                  ( ( s @ ( fun @ D @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ H ) ) ) @ ( s @ ( fun @ A @ B ) @ I ) ) )
                  = ( s @ ( fun @ D @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ H ) ) ) @ ( s @ ( fun @ A @ B ) @ I ) ) ) )
             => ! [H: $i] :
                  ( ! [I: $i,J: $i,K: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ I ) ) ) @ ( s @ ( fun @ A @ B ) @ J ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ K ) ) ) )
                    <=> ! [L: $i,M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ K ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ J ) ) ) @ ( s @ C @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) ) )
                 => ! [I: $i] :
                      ( ! [J: $i,K: $i] :
                          ( ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ J ) ) ) @ ( s @ ( fun @ A @ B ) @ K ) ) )
                          = ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ J ) ) ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) ) ) )
                     => ! [J: $i] :
                          ( ! [K: $i,L: $i] :
                              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) ) @ J ) @ ( s @ ( fun @ D @ A ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ A ) @ L ) ) ) )
                            <=> ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ A ) @ L ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ M ) ) ) @ ( s @ D @ N ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ K ) @ ( s @ D @ N ) ) ) ) ) ) )
                         => ! [K: $i] :
                              ( ! [L: $i,M: $i,N: $i] :
                                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ A @ B ) @ M ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ N ) ) ) )
                                <=> ! [O: $i,P: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ N ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ O ) ) ) @ ( s @ D @ P ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ L ) @ ( s @ ( fun @ A @ B ) @ M ) ) ) @ ( s @ D @ P ) ) ) ) ) ) )
                             => ! [L: $i] :
                                  ( ! [M: $i,N: $i] :
                                      ( ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ A @ B ) @ N ) ) )
                                      = ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ A @ B ) @ N ) ) ) ) ) )
                                 => ! [M: $i,N: $i,O: $i,P: $i] :
                                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ N ) ) ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) @ ( fun @ ( prod @ C @ D ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) ) @ J ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ P ) ) ) ) ) )
                                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ P ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_LAMBDA) ).

thf(8360,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i,I: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) @ ( s @ D @ H ) ) ) @ ( s @ C @ I ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ H ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i] :
              ( ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ H ) ) ) @ ( s @ D @ I ) ) )
              = ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ B ) @ H ) ) ) @ ( s @ D @ I ) ) ) )
         => ! [G: $i] :
              ( ! [H: $i,I: $i] :
                  ( ( s @ ( fun @ D @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ H ) ) ) @ ( s @ ( fun @ A @ B ) @ I ) ) )
                  = ( s @ ( fun @ D @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ H ) ) ) @ ( s @ ( fun @ A @ B ) @ I ) ) ) )
             => ! [H: $i] :
                  ( ! [I: $i,J: $i,K: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ I ) ) ) @ ( s @ ( fun @ A @ B ) @ J ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ K ) ) ) )
                       => ! [L: $i,M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ K ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ J ) ) ) @ ( s @ C @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) ) )
                      & ( ! [L: $i,M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ K ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ I ) @ ( s @ ( fun @ A @ B ) @ J ) ) ) @ ( s @ C @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ I ) ) ) @ ( s @ ( fun @ A @ B ) @ J ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ K ) ) ) ) ) )
                 => ! [I: $i] :
                      ( ! [J: $i,K: $i] :
                          ( ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ J ) ) ) @ ( s @ ( fun @ A @ B ) @ K ) ) )
                          = ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ J ) ) ) @ ( s @ ( fun @ A @ B ) @ K ) ) ) ) ) )
                     => ! [J: $i] :
                          ( ! [K: $i,L: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) ) @ J ) @ ( s @ ( fun @ D @ A ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ A ) @ L ) ) ) )
                               => ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ A ) @ L ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ M ) ) ) @ ( s @ D @ N ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ K ) @ ( s @ D @ N ) ) ) ) ) ) )
                              & ( ! [M: $i,N: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ A ) @ L ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ M ) ) ) @ ( s @ D @ N ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ K ) @ ( s @ D @ N ) ) ) ) ) )
                               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) ) @ J ) @ ( s @ ( fun @ D @ A ) @ K ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ A ) @ L ) ) ) ) ) )
                         => ! [K: $i] :
                              ( ! [L: $i,M: $i,N: $i] :
                                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ A @ B ) @ M ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ N ) ) ) )
                                   => ! [O: $i,P: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ N ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ O ) ) ) @ ( s @ D @ P ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ L ) @ ( s @ ( fun @ A @ B ) @ M ) ) ) @ ( s @ D @ P ) ) ) ) ) ) )
                                  & ( ! [O: $i,P: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ N ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ O ) ) ) @ ( s @ D @ P ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ L ) @ ( s @ ( fun @ A @ B ) @ M ) ) ) @ ( s @ D @ P ) ) ) ) ) )
                                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ A @ B ) @ M ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ N ) ) ) ) ) )
                             => ! [L: $i] :
                                  ( ! [M: $i,N: $i] :
                                      ( ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ A @ B ) @ N ) ) )
                                      = ( s @ ( fun @ ( prod @ C @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ bool ) @ bool ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ A @ B ) @ N ) ) ) ) ) )
                                 => ! [M: $i,N: $i,O: $i,P: $i] :
                                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ N ) ) ) ) ) @ ( s @ ( fun @ ( prod @ C @ D ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) @ ( fun @ ( prod @ C @ D ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( prod @ C @ D ) @ A ) @ bool ) ) @ J ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( prod @ C @ D ) @ bool ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ P ) ) ) ) ) )
                                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ D @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ D @ ( fun @ C @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ P ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[867]) ).

thf(682,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_ADDu_DISTRIB) ).

thf(7239,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[682]) ).

thf(950,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_NEGL) ).

thf(8830,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[950]) ).

thf(620,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
      <=> ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INFu_LTu_FINITE) ).

thf(6928,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) )
         => ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
        & ( ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).

thf(3024,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_TRIANGLE) ).

thf(20888,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3024]) ).

thf(1842,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ C ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( cart @ A @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ C ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) ) )
          = ( s @ ( cart @ A @ B ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLAMBDAu_ETA) ).

thf(14447,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ C ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) @ ( s @ num @ E ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( cart @ A @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ C ) @ ( s @ ( cart @ A @ B ) @ D ) ) ) ) )
          = ( s @ ( cart @ A @ B ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1842]) ).

thf(2313,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ F ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) ) )
     => ( ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ D )
        = ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADJOINTu_UNIQUE) ).

thf(16998,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) )
        & ! [E: $i,F: $i] :
            ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ F ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) ) )
     => ( ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ D )
        = ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2313]) ).

thf(2623,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_OVERu_UNION) ).

thf(18762,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2623]) ).

thf(55,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_LE) ).

thf(3478,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(45,axiom,
    ! [A: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ s0 )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ s0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ s0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_INF) ).

thf(3433,plain,
    ! [A: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ s0 )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ s0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ s0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(2181,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWOSETu_TRANSu_LE) ).

thf(16369,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2181]) ).

thf(242,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ ( fun @ A @ num ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ H ) ) ) )
            <=> ? [I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ J ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ G ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ A @ I ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ H ) ) ) @ ( s @ B @ I ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ ( fun @ A @ num ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) )
             => ! [F: $i] :
                  ( ! [G: $i,H: $i,I: $i] :
                      ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ ( fun @ B @ num ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) )
                      = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) )
                 => ! [G: $i] :
                      ( ! [H: $i,I: $i,J: $i,K: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                        <=> ? [L: $i,M: $i] :
                              ( ( ( p @ ( s @ bool @ M ) )
                              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ J ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ K ) ) ) @ ( s @ bool @ M ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                     => ! [H: $i] :
                          ( ! [I: $i,J: $i,K: $i,L: $i] :
                              ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) ) @ H ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ K ) ) ) @ ( s @ A @ L ) ) )
                              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ A @ L ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ ( fun @ B @ num ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ K ) ) ) @ ( s @ A @ L ) ) ) ) ) )
                         => ! [I: $i,J: $i,K: $i,L: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) ) )
                             => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) ) @ H ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ J ) ) ) ) )
                                = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ J ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_NSUMu_RESTRICT) ).

thf(4548,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ ( fun @ A @ num ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ H ) ) ) )
               => ? [I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ A @ I ) ) ) ) ) )
              & ( ? [I: $i,J: $i] :
                    ( ( ( p @ ( s @ bool @ J ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) @ ( s @ B @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ J ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ A @ I ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ A @ H ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ H ) ) ) @ ( s @ B @ I ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ ( fun @ A @ num ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) )
             => ! [F: $i] :
                  ( ! [G: $i,H: $i,I: $i] :
                      ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ ( fun @ B @ num ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) )
                      = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) )
                 => ! [G: $i] :
                      ( ! [H: $i,I: $i,J: $i,K: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) )
                           => ? [L: $i,M: $i] :
                                ( ( ( p @ ( s @ bool @ M ) )
                                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ J ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ J ) ) ) @ ( s @ B @ L ) ) ) ) )
                                 => ( p @ ( s @ bool @ M ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ K ) ) ) @ ( s @ bool @ M ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                          & ( ? [L: $i,M: $i] :
                                ( ( ( p @ ( s @ bool @ M ) )
                                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ J ) ) ) @ ( s @ B @ L ) ) ) ) ) )
                                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) @ ( s @ A @ J ) ) ) @ ( s @ B @ L ) ) ) ) )
                                 => ( p @ ( s @ bool @ M ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ K ) ) ) @ ( s @ bool @ M ) ) ) @ ( s @ B @ L ) ) ) ) )
                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) )
                     => ! [H: $i] :
                          ( ! [I: $i,J: $i,K: $i,L: $i] :
                              ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) ) @ H ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ K ) ) ) @ ( s @ A @ L ) ) )
                              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ B @ bool ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ J ) ) ) @ ( s @ A @ L ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ ( fun @ B @ num ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ K ) ) ) @ ( s @ A @ L ) ) ) ) ) )
                         => ! [I: $i,J: $i,K: $i,L: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) ) )
                             => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) ) @ H ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ J ) ) ) ) )
                                = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ L ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ K ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ J ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(979,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ E ) )
      <=> ( ( ( s @ int @ C )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ int @ D )
            = ( s @ int @ A ) ) ) )
      & ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
          <=> ( ( s @ int @ B )
              = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
          & ? [G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ D ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) )
                  & ( ( s @ int @ A )
                    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_DIVMODu_EXISTu_0) ).

thf(8973,plain,
    ! [A: $i,B: $i] :
    ? [C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ E ) )
       => ( ( ( s @ int @ C )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ int @ D )
            = ( s @ int @ A ) ) ) )
      & ( ( ( ( s @ int @ C )
            = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ( ( s @ int @ D )
            = ( s @ int @ A ) ) )
       => ( p @ ( s @ bool @ E ) ) )
      & ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
           => ( ( s @ int @ B )
              = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
          & ( ( ( s @ int @ B )
              = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
           => ( p @ ( s @ bool @ F ) ) )
          & ? [G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ D ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) )
                  & ( ( s @ int @ A )
                    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ D ) ) ) ) ) )
              & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ D ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) )
                  & ( ( s @ int @ A )
                    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ D ) ) ) ) )
               => ( p @ ( s @ bool @ G ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[979]) ).

thf(3099,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
       => ( p @ ( s @ bool @ C ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_IMPu_FORALLu_THM) ).

thf(21359,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ C ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ C ) ) )
       => ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3099]) ).

thf(194,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_2) ).

thf(4278,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(1900,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ g_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
    <=> ? [E: $i] :
        ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ? [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
              & ( ( s @ A @ F )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aGEu_C) ).

thf(14732,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ g_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
       => ? [E: $i] :
          ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( ( s @ A @ F )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) ) )
      & ( ? [E: $i] :
          ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
                & ( ( s @ A @ F )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ g_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1900]) ).

thf(375,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) )
        = ( s @ real @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_RINVu_UNIQ) ).

thf(5490,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) )
        = ( s @ real @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).

thf(1988,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( ( s @ real @ A )
         != ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_LE) ).

thf(15347,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
          & ( ( s @ real @ A )
           != ( s @ real @ B ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
          & ( ( s @ real @ A )
           != ( s @ real @ B ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1988]) ).

thf(2190,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
          & ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ F ) ) ) ) )
             => ( ( s @ B @ E )
                = ( s @ B @ F ) ) ) )
    <=> ? [D: $i] :
        ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) )
        <=> ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) )
            = ( s @ B @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIQUEu_SKOLEMu_ALT) ).

thf(16418,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
            & ! [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ F ) ) ) ) )
               => ( ( s @ B @ E )
                  = ( s @ B @ F ) ) ) )
       => ? [D: $i] :
          ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) )
                = ( s @ B @ F ) ) )
            & ( ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) )
                = ( s @ B @ F ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) ) )
      & ( ? [D: $i] :
          ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) )
                = ( s @ B @ F ) ) )
            & ( ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) )
                = ( s @ B @ F ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
       => ! [D: $i] :
            ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
            & ! [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ B @ F ) ) ) ) )
               => ( ( s @ B @ E )
                  = ( s @ B @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2190]) ).

thf(1819,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_COUNTABLEu_CONG) ).

thf(14306,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1819]) ).

thf(51,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ B )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ARCH) ).

thf(3457,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ B )
       != ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(1937,axiom,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
           => ( ( s @ A @ C )
              = ( s @ A @ D ) ) ) )
    <=> ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
           => ( ( s @ A @ C )
              = ( s @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_UNIQUEu_THM) ).

thf(15056,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1937]) ).

thf(258,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum )
      = ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) ) @ iterate ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ansum) ).

thf(4698,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum )
      = ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ num ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) ) @ iterate ) @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(627,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIMINDEXu_GEu_1) ).

thf(6965,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[627]) ).

thf(1889,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_SUB2) ).

thf(14678,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1889]) ).

thf(2136,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ B ) @ A ) @ ( cart @ A @ B ) ) @ mku_cart ) @ ( s @ ( fun @ ( finite_image @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ ( finite_image @ B ) @ A ) ) @ destu_cart ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) ) )
      = ( s @ ( cart @ A @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',acartu_tybiju_conjunct0) ).

thf(16121,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ B ) @ A ) @ ( cart @ A @ B ) ) @ mku_cart ) @ ( s @ ( fun @ ( finite_image @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ ( finite_image @ B ) @ A ) ) @ destu_cart ) @ ( s @ ( cart @ A @ B ) @ C ) ) ) ) )
      = ( s @ ( cart @ A @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2136]) ).

thf(1151,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( ( s @ A @ F )
                   != ( s @ A @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDELETE) ).

thf(9912,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ A @ F )
                     != ( s @ A @ D ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ A @ F )
                     != ( s @ A @ D ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ A @ F )
                     != ( s @ A @ D ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ A @ F )
                     != ( s @ A @ D ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1151]) ).

thf(1645,axiom,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ x ) ) ) @ ( s @ nadd @ y ) ) )
    = ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ x ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ y ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ahrealu_leu_th) ).

thf(13211,plain,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ x ) ) ) @ ( s @ nadd @ y ) ) )
    = ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ x ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ y ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1645]) ).

thf(1159,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_OFu_NUMu_ADD) ).

thf(9966,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1159]) ).

thf(406,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_LMUL) ).

thf(5653,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).

thf(1304,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_OFFSET) ).

thf(10806,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ n ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ n ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1304]) ).

thf(2402,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_SYM) ).

thf(17501,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2402]) ).

thf(269,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_RESTRICT) ).

thf(4770,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(2655,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_SUBSET) ).

thf(18894,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2655]) ).

thf(770,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EVENu_conjunct3) ).

thf(7764,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[770]) ).

thf(2518,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_RULESu_conjunct0) ).

thf(18149,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2518]) ).

thf(1047,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
      & ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_0) ).

thf(9306,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
      & ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1047]) ).

thf(2365,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ B ) ) ) ) ) )
    <=> ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_DELETE) ).

thf(17259,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ B ) ) ) ) ) )
       => ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2365]) ).

thf(2534,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_ABS) ).

thf(18281,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2534]) ).

thf(1570,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_01) ).

thf(12690,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1570]) ).

thf(2390,axiom,
    ! [A: $i] :
      ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ s0 )
        = ( s @ ( fun @ A @ bool ) @ univ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_UNIV) ).

thf(17391,plain,
    ! [A: $i] :
      ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ s0 )
          = ( s @ ( fun @ A @ bool ) @ univ ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ s0 )
          = ( s @ ( fun @ A @ bool ) @ univ ) )
       => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2390]) ).

thf(1840,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) )
    <=> ( ! [C: $i,D: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
        & ! [C: $i,D: $i,E: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ E ) ) ) )
        & ! [C: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) ) ) @ ( s @ A @ C ) ) )
            = ( s @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amonoidal) ).

thf(14432,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) )
       => ( ! [C: $i,D: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
          & ! [C: $i,D: $i,E: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ E ) ) ) )
          & ! [C: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) ) ) @ ( s @ A @ C ) ) )
              = ( s @ A @ C ) ) ) )
      & ( ( ! [C: $i,D: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
          & ! [C: $i,D: $i,E: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ A @ E ) ) ) )
          & ! [C: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) ) ) @ ( s @ A @ C ) ) )
              = ( s @ A @ C ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1840]) ).

thf(2741,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) )
       != ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aODDu_ADD) ).

thf(19324,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) )
         != ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ B ) ) ) ) )
      & ( ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) )
         != ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2741]) ).

thf(2017,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_LE) ).

thf(15525,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2017]) ).

thf(792,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
        = ( s @ real @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPOWu_2u_SQRT) ).

thf(7904,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
        = ( s @ real @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[792]) ).

thf(382,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_PRESUC) ).

thf(5536,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).

thf(2338,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_CLAUSESu_conjunct8) ).

thf(17121,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2338]) ).

thf(2764,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ B ) ) )
        = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ C ) ) ) )
    <=> ( ( s @ num @ B )
        = ( s @ num @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJNu_INJ) ).

thf(19456,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ B ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ C ) ) ) )
       => ( ( s @ num @ B )
          = ( s @ num @ C ) ) )
      & ( ( ( s @ num @ B )
          = ( s @ num @ C ) )
       => ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ B ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2764]) ).

thf(584,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ univ )
      = ( s @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( finite_sum @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( finite_sum @ A @ B ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( finite_sum @ A @ B ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ ( finite_sum @ A @ B ) ) @ mku_finiteu_sum ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_SUMu_IMAGE) ).

thf(6721,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ univ )
      = ( s @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( finite_sum @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( finite_sum @ A @ B ) ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( finite_sum @ A @ B ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ ( finite_sum @ A @ B ) ) @ mku_finiteu_sum ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).

thf(2921,axiom,
    ( ( s @ bool @ t )
    = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTu_DEF) ).

thf(20346,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2921]) ).

thf(2269,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
         => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) ) )
    <=> ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
        = ( s @ ( fun @ B @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPPORTu_EMPTY) ).

thf(16814,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) ) )
       => ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
          = ( s @ ( fun @ B @ bool ) @ empty ) ) )
      & ( ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) )
          = ( s @ ( fun @ B @ bool ) @ empty ) )
       => ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2269]) ).

thf(220,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i,C: $i] :
        ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LINVu_LEMMA5) ).

thf(4419,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i,C: $i] :
        ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(2834,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
        | ? [C: $i,D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
            & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORDINALu_UP) ).

thf(19876,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ C ) ) ) )
        | ? [C: $i,D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
            & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2834]) ).

thf(1376,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ E ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_LEu_NUMSEG) ).

thf(11450,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ E ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1376]) ).

thf(1652,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_ADDu_LID) ).

thf(13271,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1652]) ).

thf(457,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SGNu_EQu_conjunct2) ).

thf(5966,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).

thf(3020,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LDISTRIB) ).

thf(20874,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3020]) ).

thf(975,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABS) ).

thf(8962,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ int @ ( fun @ int @ int ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[975]) ).

thf(1300,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_01) ).

thf(10796,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1300]) ).

thf(973,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) ) )
     => ? [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_ASu_IMAGE) ).

thf(8957,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) ) )
     => ? [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[973]) ).

thf(603,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_RINV) ).

thf(6835,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ B ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).

thf(2023,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( sum0 @ B @ A ) @ B ) @ outl ) @ ( s @ ( sum0 @ B @ A ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ B @ A ) ) @ inl ) @ ( s @ B @ x ) ) ) ) )
      = ( s @ B @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aOUTL) ).

thf(15548,plain,
    ! [A: $i,B: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( sum0 @ B @ A ) @ B ) @ outl ) @ ( s @ ( sum0 @ B @ A ) @ ( i @ ( s @ ( fun @ B @ ( sum0 @ B @ A ) ) @ inl ) @ ( s @ B @ x ) ) ) ) )
      = ( s @ B @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2023]) ).

thf(2552,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMu_REPu_RULESu_conjunct1) ).

thf(18355,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ ( i @ ( s @ ( fun @ ind0 @ ind0 ) @ indu_suc ) @ ( s @ ind0 @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2552]) ).

thf(949,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_BOUNDu_LTu_ALL) ).

thf(8825,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ empty ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[949]) ).

thf(2838,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ n10 ) @ B )
      = ( s @ ( fun @ A @ n10 ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aoneu_axiom) ).

thf(19897,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ n10 ) @ B )
      = ( s @ ( fun @ A @ n10 ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2838]) ).

thf(147,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_LZERO) ).

thf(3928,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(189,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
    <=> ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
          <=> ( ( ( s @ num @ B )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            <=> ( ( s @ num @ C )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          & ? [E: $i] :
              ( ( ( p @ ( s @ bool @ E ) )
              <=> ( ( s @ num @ A )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
              & ? [F: $i] :
                  ( ( ( p @ ( s @ bool @ F ) )
                  <=> ( ( ( s @ num @ A )
                        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
                      | ( ( s @ num @ B )
                        = ( s @ num @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_EXP) ).

thf(4185,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
       => ? [D: $i] :
            ( ( ( p @ ( s @ bool @ D ) )
             => ( ( ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( ( s @ num @ C )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                & ( ( ( s @ num @ C )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
            & ( ( ( ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( ( s @ num @ C )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                & ( ( ( s @ num @ C )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
             => ( p @ ( s @ bool @ D ) ) )
            & ? [E: $i] :
                ( ( ( p @ ( s @ bool @ E ) )
                 => ( ( s @ num @ A )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                & ( ( ( s @ num @ A )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( p @ ( s @ bool @ E ) ) )
                & ? [F: $i] :
                    ( ( ( p @ ( s @ bool @ F ) )
                     => ( ( ( s @ num @ A )
                          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
                        | ( ( s @ num @ B )
                          = ( s @ num @ C ) ) ) )
                    & ( ( ( ( s @ num @ A )
                          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
                        | ( ( s @ num @ B )
                          = ( s @ num @ C ) ) )
                     => ( p @ ( s @ bool @ F ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ F ) ) ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( ( p @ ( s @ bool @ D ) )
             => ( ( ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( ( s @ num @ C )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                & ( ( ( s @ num @ C )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
            & ( ( ( ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( ( s @ num @ C )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                & ( ( ( s @ num @ C )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
             => ( p @ ( s @ bool @ D ) ) )
            & ? [E: $i] :
                ( ( ( p @ ( s @ bool @ E ) )
                 => ( ( s @ num @ A )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                & ( ( ( s @ num @ A )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( p @ ( s @ bool @ E ) ) )
                & ? [F: $i] :
                    ( ( ( p @ ( s @ bool @ F ) )
                     => ( ( ( s @ num @ A )
                          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
                        | ( ( s @ num @ B )
                          = ( s @ num @ C ) ) ) )
                    & ( ( ( ( s @ num @ A )
                          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
                        | ( ( s @ num @ B )
                          = ( s @ num @ C ) ) )
                     => ( p @ ( s @ bool @ F ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ F ) ) ) ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(2238,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_NEGu_COMPONENT) ).

thf(16663,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2238]) ).

thf(1826,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_LNEG) ).

thf(14344,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1826]) ).

thf(1537,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_CLAUSESu_conjunct0) ).

thf(12484,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1537]) ).

thf(1541,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ card ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) )
          = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_NUMSEGu_LT) ).

thf(12495,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ card ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) )
          = ( s @ num @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1541]) ).

thf(2742,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONDu_RATOR) ).

thf(19331,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2742]) ).

thf(1972,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MINu_LT) ).

thf(15260,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1972]) ).

thf(2100,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
     => ( ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
           => ( ( s @ B @ C )
              = ( s @ B @ D ) ) )
      <=> ? [C: $i] :
            ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) )
            & ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
               => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
                  = ( s @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJECTIVEu_LEFTu_INVERSEu_NONEMPTY) ).

thf(15903,plain,
    ! [A: $i,B: $i] :
      ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
     => ( ( ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
             => ( ( s @ B @ C )
                = ( s @ B @ D ) ) )
         => ? [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                 => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
                    = ( s @ B @ D ) ) ) ) )
        & ( ? [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                 => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
                    = ( s @ B @ D ) ) ) )
         => ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
             => ( ( s @ B @ C )
                = ( s @ B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2100]) ).

thf(1976,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aETAu_AX) ).

thf(15277,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1976]) ).

thf(2657,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ! [B: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_REALu_INTERVALu_conjunct0) ).

thf(18900,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ real @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) @ ( s @ real @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ! [B: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2657]) ).

thf(843,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ B @ int ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ int ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
        & ! [E: $i,F: $i] :
            ( ! [G: $i,H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ B @ int ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ int ) ) @ D ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ B @ int ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ int ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
     => ! [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_INTu_MEASUREu_2) ).

thf(8228,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ B @ int ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ int ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
        & ! [E: $i,F: $i] :
            ( ! [G: $i,H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ B @ int ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ int ) ) @ D ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ B @ int ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ int ) ) @ D ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
     => ! [E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[843]) ).

thf(610,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_TRIANGLEu_HALFu_L) ).

thf(6867,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ e0 ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ e0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).

thf(1309,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ D ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) )
     => ! [E: $i] :
          ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ B ) ) @ fstcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) )
          = ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( cart @ A @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ D ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',afstcart) ).

thf(10905,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ D ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) )
     => ! [E: $i] :
          ( ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ B ) ) @ fstcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) )
          = ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( cart @ A @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( fun @ num @ A ) ) @ D ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1309]) ).

thf(1716,axiom,
    ? [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        <=> ? [C: $i] :
              ( ( s @ real @ B )
              = ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ A ) @ ( s @ hreal @ C ) ) ) ) )
      & ! [B: $i,C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ A ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ A ) @ ( s @ hreal @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_HREALu_LEMMA1) ).

thf(13615,plain,
    ? [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
           => ? [C: $i] :
                ( ( s @ real @ B )
                = ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ A ) @ ( s @ hreal @ C ) ) ) ) )
          & ( ? [C: $i] :
                ( ( s @ real @ B )
                = ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ A ) @ ( s @ hreal @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) )
      & ! [B: $i,C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ A ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ hreal @ real ) @ A ) @ ( s @ hreal @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1716]) ).

thf(2633,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIRWISEu_MONO) ).

thf(18794,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2633]) ).

thf(2396,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_SUB) ).

thf(17479,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_sub ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2396]) ).

thf(3068,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_CASES) ).

thf(21185,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3068]) ).

thf(324,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ D ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) ) ) )
        <=> ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ p0 ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ ( fun @ ( prod @ B @ C ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ D ) ) ) @ ( s @ ( prod @ B @ C ) @ E ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ p0 ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ B ) @ fst ) @ ( s @ ( prod @ B @ C ) @ E ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ C ) @ snd ) @ ( s @ ( prod @ B @ C ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLAMBDAu_PAIR) ).

thf(5124,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ D ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) ) ) )
           => ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ p0 ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ) ) ) )
          & ( ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ p0 ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ D ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ E ) ) ) ) ) )
     => ! [E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ ( fun @ ( prod @ B @ C ) @ A ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ bool ) @ D ) ) ) @ ( s @ ( prod @ B @ C ) @ E ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ p0 ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ B ) @ fst ) @ ( s @ ( prod @ B @ C ) @ E ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ C ) @ snd ) @ ( s @ ( prod @ B @ C ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).

thf(355,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_MULu_SYMu_EQ) ).

thf(5332,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) )
      = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).

thf(1904,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ D ) ) )
     != ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZCONSTRu_ZBOT) ).

thf(14802,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ D ) ) )
     != ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1904]) ).

thf(1087,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_MULu_EQu_conjunct0) ).

thf(9515,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1087]) ).

thf(2926,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) ) @ ( s @ bool @ C ) ) ) )
    <=> ( ( ~ ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ B ) ) )
        & ( ( p @ ( s @ bool @ A ) )
          | ( p @ ( s @ bool @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONDu_EXPAND) ).

thf(20372,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) ) @ ( s @ bool @ C ) ) ) )
       => ( ( ~ ( p @ ( s @ bool @ A ) )
            | ( p @ ( s @ bool @ B ) ) )
          & ( ( p @ ( s @ bool @ A ) )
            | ( p @ ( s @ bool @ C ) ) ) ) )
      & ( ( ( ~ ( p @ ( s @ bool @ A ) )
            | ( p @ ( s @ bool @ B ) ) )
          & ( ( p @ ( s @ bool @ A ) )
            | ( p @ ( s @ bool @ C ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) ) @ ( s @ bool @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2926]) ).

thf(1262,axiom,
    ! [A: $i] :
      ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ p0 ) @ ( s @ ( finite_image @ A ) @ B ) ) ) )
    <=> ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ p0 ) @ ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_FINITEu_INDEX) ).

thf(10578,plain,
    ! [A: $i] :
      ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ p0 ) @ ( s @ ( finite_image @ A ) @ B ) ) ) )
       => ! [B: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ p0 ) @ ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ B ) ) ) ) ) ) ) )
      & ( ! [B: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ p0 ) @ ( s @ ( finite_image @ A ) @ ( i @ ( s @ ( fun @ num @ ( finite_image @ A ) ) @ finiteu_index ) @ ( s @ num @ B ) ) ) ) ) ) )
       => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ p0 ) @ ( s @ ( finite_image @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1262]) ).

thf(2875,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
    <=> ( ! [D: $i,E: $i] :
            ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) )
        & ! [D: $i,E: $i] :
            ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alinear) ).

thf(20126,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
       => ( ! [D: $i,E: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) )
          & ! [D: $i,E: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ) )
      & ( ( ! [D: $i,E: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) )
          & ! [D: $i,E: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2875]) ).

thf(2330,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( option @ A ) @ bool ) @ B ) @ ( s @ ( option @ A ) @ none ) ) ) )
        & ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( option @ A ) @ bool ) @ B ) @ ( s @ ( option @ A ) @ ( i @ ( s @ ( fun @ A @ ( option @ A ) ) @ some ) @ ( s @ A @ C ) ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( option @ A ) @ bool ) @ B ) @ ( s @ ( option @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aoptionu_elim) ).

thf(17088,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( option @ A ) @ bool ) @ B ) @ ( s @ ( option @ A ) @ none ) ) ) )
        & ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( option @ A ) @ bool ) @ B ) @ ( s @ ( option @ A ) @ ( i @ ( s @ ( fun @ A @ ( option @ A ) ) @ some ) @ ( s @ A @ C ) ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( option @ A ) @ bool ) @ B ) @ ( s @ ( option @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2330]) ).

thf(120,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GTu_conjunct5) ).

thf(3797,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(1705,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) )
    <=> ( ( ( s @ real @ B )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        | ( ( s @ ( cart @ real @ A ) @ C )
          = ( s @ ( cart @ real @ A ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_LCANCEL) ).

thf(13549,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) )
       => ( ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ ( cart @ real @ A ) @ C )
            = ( s @ ( cart @ real @ A ) @ D ) ) ) )
      & ( ( ( ( s @ real @ B )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
          | ( ( s @ ( cart @ real @ A ) @ C )
            = ( s @ ( cart @ real @ A ) @ D ) ) )
       => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1705]) ).

thf(1532,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_LNEG) ).

thf(12452,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1532]) ).

thf(1514,axiom,
    ( ! [A: $i] :
        ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ n10 @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ n10 @ bool ) @ univ ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
  <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_DIMINDEXu_1) ).

thf(12302,plain,
    ( ( ! [A: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ n10 @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ n10 @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
    & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [A: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ n10 @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ n10 @ bool ) @ univ ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1514]) ).

thf(2765,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_IMAGEu_SUBSET) ).

thf(19463,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ A ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2765]) ).

thf(2899,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_LE) ).

thf(20234,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2899]) ).

thf(1565,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) ) @ A ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) )
        <=> ( ( s @ hreal @ B )
            = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) )
     => ! [B: $i,C: $i] :
        ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
          <=> ( ( s @ hreal @ C )
              = ( s @ hreal @ B ) ) )
          & ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) )
            = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ hreal @ bool ) @ hreal ) @ h_ ) @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) ) @ A ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ hreal @ bool ) @ hreal ) @ h_ ) @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) ) @ A ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',atrealu_inv) ).

thf(12662,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) ) @ A ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) )
           => ( ( s @ hreal @ B )
              = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) )
          & ( ( ( s @ hreal @ B )
              = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) ) @ A ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
        ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
           => ( ( s @ hreal @ C )
              = ( s @ hreal @ B ) ) )
          & ( ( ( s @ hreal @ C )
              = ( s @ hreal @ B ) )
           => ( p @ ( s @ bool @ D ) ) )
          & ( ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) )
            = ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ hreal @ bool ) @ hreal ) @ h_ ) @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) ) @ A ) @ ( s @ hreal @ C ) ) ) @ ( s @ hreal @ B ) ) ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( prod @ hreal @ hreal ) ) ) @ c_ ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ hreal @ bool ) @ hreal ) @ h_ ) @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) ) @ A ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1565]) ).

thf(2272,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_NOTu_LT) ).

thf(16835,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2272]) ).

thf(747,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_UNIONu_LZERO) ).

thf(7670,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[747]) ).

thf(344,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_RNEG) ).

thf(5258,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).

thf(2946,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LEu_conjunct6) ).

thf(20487,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2946]) ).

thf(54,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
        = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) )
    <=> ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
         => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_EQUALu_ROWS) ).

thf(3465,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
          = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) )
       => ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
           => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) )
      & ( ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
           => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ A ) ) ) @ row ) @ ( s @ num @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
       => ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C )
          = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(2106,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_IMPu_LE) ).

thf(15964,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2106]) ).

thf(2170,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_RMUL) ).

thf(16320,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2170]) ).

thf(100,axiom,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( list @ A ) @ nil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFILTERu_conjunct0) ).

thf(3729,plain,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ filter0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( list @ A ) @ nil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(1456,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_MULu_VSUM) ).

thf(11963,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ real @ B ) ) ) @ column ) @ ( s @ num @ F ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1456]) ).

thf(180,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_DIFFS) ).

thf(4121,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(613,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_MONOu_LE) ).

thf(6903,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).

thf(2580,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_sub) ).

thf(18495,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2580]) ).

thf(2850,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) ) ) @ a_a_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',acong) ).

thf(19961,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) ) ) @ a_a_ ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2850]) ).

thf(2294,axiom,
    ( ( s @ bool @ u_falsityu_ )
    = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',au_FALSITYu_) ).

thf(16931,plain,
    ( ( s @ bool @ u_falsityu_ )
    = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2294]) ).

thf(2199,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_EMPTY) ).

thf(16470,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2199]) ).

thf(507,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ARCHu_LEMMA) ).

thf(6230,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ nadd @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).

thf(2773,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_0) ).

thf(19536,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2773]) ).

thf(520,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_MONO) ).

thf(6322,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(2457,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( option @ B ) @ A ) @ E ) @ ( s @ ( option @ B ) @ none ) ) )
        = ( s @ A @ C ) )
      & ! [F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( option @ B ) @ A ) @ E ) @ ( s @ ( option @ B ) @ ( i @ ( s @ ( fun @ B @ ( option @ B ) ) @ some ) @ ( s @ B @ F ) ) ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aoptionu_RECURSION) ).

thf(17842,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( option @ B ) @ A ) @ E ) @ ( s @ ( option @ B ) @ none ) ) )
        = ( s @ A @ C ) )
      & ! [F: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( option @ B ) @ A ) @ E ) @ ( s @ ( option @ B ) @ ( i @ ( s @ ( fun @ B @ ( option @ B ) ) @ some ) @ ( s @ B @ F ) ) ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2457]) ).

thf(1395,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ D ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_CONST) ).

thf(11584,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ D ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) )
            = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1395]) ).

thf(1663,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_DIFFSQ) ).

thf(13303,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1663]) ).

thf(1797,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ B @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ B ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ E @ A ) @ I ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ J ) ) ) @ ( s @ C @ K ) ) ) @ ( s @ A @ L ) ) ) )
        <=> ? [M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ u_unguardedu_pattern ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ G ) @ ( s @ D @ M ) ) ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ I ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ J ) @ ( s @ C @ K ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ A @ L ) ) ) ) ) ) )
     => ! [G: $i] :
          ( ! [H: $i,I: $i,J: $i,K: $i,L: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ D @ B ) @ H ) ) ) @ ( s @ ( fun @ E @ A ) @ I ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ J ) ) ) @ ( s @ C @ K ) ) ) @ ( s @ B @ L ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ B @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ B ) @ H ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ I ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ J ) ) ) @ ( s @ C @ K ) ) ) )
         => ! [H: $i] :
              ( ! [I: $i,J: $i,K: $i,L: $i,M: $i] :
                  ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ E @ A ) @ K ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ C @ M ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ A ) ) @ u_match ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ I ) @ ( s @ C @ M ) ) ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ E @ A ) @ K ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) )
             => ! [I: $i] :
                  ( ! [J: $i,K: $i,L: $i,M: $i] :
                      ( ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) @ I ) @ ( s @ ( fun @ C @ B ) @ J ) ) ) @ ( s @ ( fun @ D @ B ) @ K ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) )
                      = ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ J ) ) ) @ ( s @ ( fun @ D @ B ) @ K ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ L ) ) ) )
                 => ! [J: $i,K: $i,L: $i,M: $i,N: $i,O: $i] :
                      ( ( ! [P: $i,Q: $i,R: $i,S: $i] :
                            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ P ) ) ) @ ( s @ C @ Q ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ R ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ Q ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ S ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ Q ) ) ) ) )
                           => ( ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ O ) @ ( s @ C @ Q ) ) ) @ ( s @ D @ R ) ) )
                              = ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ O ) @ ( s @ C @ Q ) ) ) @ ( s @ D @ S ) ) ) ) )
                        & ! [P: $i,Q: $i,R: $i] :
                            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ P ) ) ) @ ( s @ C @ Q ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ R ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ Q ) ) ) ) )
                           => ! [S: $i] :
                                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) @ ( s @ E @ S ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ O ) @ ( s @ C @ Q ) ) ) @ ( s @ D @ R ) ) ) ) ) )
                               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) @ ( s @ E @ S ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ C @ E ) @ L ) @ ( s @ C @ Q ) ) ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ C @ E ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) @ I ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ ( fun @ D @ B ) @ N ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ O ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPERADMISSIBLEu_MATCHu_UNGUARDEDu_PATTERN) ).

thf(14150,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ B @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ B ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ E @ A ) @ I ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ J ) ) ) @ ( s @ C @ K ) ) ) @ ( s @ A @ L ) ) ) )
           => ? [M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ u_unguardedu_pattern ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ G ) @ ( s @ D @ M ) ) ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ I ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ J ) @ ( s @ C @ K ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ A @ L ) ) ) ) ) ) )
          & ( ? [M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ u_unguardedu_pattern ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ G ) @ ( s @ D @ M ) ) ) ) ) @ ( s @ B @ H ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ geq ) @ ( s @ A @ ( i @ ( s @ ( fun @ E @ A ) @ I ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ J ) @ ( s @ C @ K ) ) ) @ ( s @ D @ M ) ) ) ) ) ) ) @ ( s @ A @ L ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ B @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ B ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ E @ A ) @ I ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ J ) ) ) @ ( s @ C @ K ) ) ) @ ( s @ A @ L ) ) ) ) ) )
     => ! [G: $i] :
          ( ! [H: $i,I: $i,J: $i,K: $i,L: $i] :
              ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ D @ B ) @ H ) ) ) @ ( s @ ( fun @ E @ A ) @ I ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ J ) ) ) @ ( s @ C @ K ) ) ) @ ( s @ B @ L ) ) )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ B @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ A @ bool ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ D @ B ) @ H ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ I ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ J ) ) ) @ ( s @ C @ K ) ) ) )
         => ! [H: $i] :
              ( ! [I: $i,J: $i,K: $i,L: $i,M: $i] :
                  ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ I ) ) ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ E @ A ) @ K ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ C @ M ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ A ) ) @ u_match ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ I ) @ ( s @ C @ M ) ) ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ D @ B ) @ J ) ) ) @ ( s @ ( fun @ E @ A ) @ K ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) )
             => ! [I: $i] :
                  ( ! [J: $i,K: $i,L: $i,M: $i] :
                      ( ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) @ I ) @ ( s @ ( fun @ C @ B ) @ J ) ) ) @ ( s @ ( fun @ D @ B ) @ K ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) )
                      = ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ C @ A ) ) ) ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ J ) ) ) @ ( s @ ( fun @ D @ B ) @ K ) ) ) @ ( s @ ( fun @ E @ A ) @ M ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ L ) ) ) )
                 => ! [J: $i,K: $i,L: $i,M: $i,N: $i,O: $i] :
                      ( ( ! [P: $i,Q: $i,R: $i,S: $i] :
                            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ P ) ) ) @ ( s @ C @ Q ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ R ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ Q ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ S ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ Q ) ) ) ) )
                           => ( ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ O ) @ ( s @ C @ Q ) ) ) @ ( s @ D @ R ) ) )
                              = ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ O ) @ ( s @ C @ Q ) ) ) @ ( s @ D @ S ) ) ) ) )
                        & ! [P: $i,Q: $i,R: $i] :
                            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) @ ( s @ ( fun @ E @ A ) @ P ) ) ) @ ( s @ C @ Q ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ N ) @ ( s @ D @ R ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ Q ) ) ) ) )
                           => ! [S: $i] :
                                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) @ ( s @ E @ S ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ O ) @ ( s @ C @ Q ) ) ) @ ( s @ D @ R ) ) ) ) ) )
                               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) @ ( s @ E @ S ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ C @ E ) @ L ) @ ( s @ C @ Q ) ) ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ C @ E ) @ ( fun @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ E @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ C @ E ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ C @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ C @ A ) ) ) ) ) @ I ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ ( fun @ D @ B ) @ N ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ E ) ) @ O ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1797]) ).

thf(590,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
        ? [H: $i] :
          ( ( ( p @ ( s @ bool @ H ) )
          <=> ( ( s @ B @ G )
              = ( s @ B @ D ) ) )
          & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ F ) ) ) @ ( s @ B @ G ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ F ) ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_SUPPORTu_DELTA) ).

thf(6753,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
        ? [H: $i] :
          ( ( ( p @ ( s @ bool @ H ) )
           => ( ( s @ B @ G )
              = ( s @ B @ D ) ) )
          & ( ( ( s @ B @ G )
              = ( s @ B @ D ) )
           => ( p @ ( s @ bool @ H ) ) )
          & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ F ) ) ) @ ( s @ B @ G ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ F ) ) ) ) ) ) )
     => ! [D: $i,E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ B @ A ) ) ) ) @ C ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).

thf(3035,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_SUB2) ).

thf(20959,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3035]) ).

thf(527,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
      <=> ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_INFu_FINITE) ).

thf(6341,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) )
         => ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
        & ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).

thf(57,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ D ) ) )
          = ( s @ bool @ f0 ) )
     => ! [D: $i,E: $i] :
        ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
          <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ D ) ) ) ) ) )
              & ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ D ) ) ) @ ( s @ B @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ D ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( ( s @ B @ G )
                    = ( s @ B @ H ) ) ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) ) @ u_match ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ B ) @ h_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ D ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ B ) @ h_ ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',au_MATCH) ).

thf(3484,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ D ) ) )
          = ( s @ bool @ f0 ) )
     => ! [D: $i,E: $i] :
        ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ D ) ) ) ) ) )
              & ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ D ) ) ) @ ( s @ B @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ D ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( ( s @ B @ G )
                    = ( s @ B @ H ) ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ D ) ) ) ) ) )
              & ! [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ D ) ) ) @ ( s @ B @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ D ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( ( s @ B @ G )
                    = ( s @ B @ H ) ) ) )
           => ( p @ ( s @ bool @ F ) ) )
          & ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ B ) ) @ u_match ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ B ) @ h_ ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ D ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ B ) @ h_ ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(1674,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ reverse ) @ ( s @ ( list @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAPu_REVERSE) ).

thf(13364,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ D ) ) ) ) )
      = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ reverse ) @ ( s @ ( list @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1674]) ).

thf(1123,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_RNEG) ).

thf(9802,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1123]) ).

thf(129,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_ADD) ).

thf(3825,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(576,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTEu_ADD2) ).

thf(6689,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ B ) ) ) @ ( s @ int @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).

thf(2360,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_ANDu_THM) ).

thf(17230,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          & ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2360]) ).

thf(362,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_LEu_L1) ).

thf(5382,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).

thf(2230,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
    <=> ? [D: $i,E: $i] :
          ( ( s @ ( list @ A ) @ C )
          = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMu_APPENDu_DECOMPOSE) ).

thf(16635,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) )
       => ? [D: $i,E: $i] :
            ( ( s @ ( list @ A ) @ C )
            = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) ) )
      & ( ? [D: $i,E: $i] :
            ( ( s @ ( list @ A ) @ C )
            = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2230]) ).

thf(3000,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_RDISTRIB) ).

thf(20785,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_mul ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3000]) ).

thf(646,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_CLAUSESu_conjunct0) ).

thf(7044,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[646]) ).

thf(3102,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
    <=> ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        | ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_ORu_THM) ).

thf(21382,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3102]) ).

thf(1942,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
            & ! [D: $i,E: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
         => ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ! [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ! [D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
             => ( ( s @ ( fun @ A @ bool ) @ C )
                = ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZLu_SUBSETS) ).

thf(15067,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
            & ! [D: $i,E: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
         => ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ! [E: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) )
     => ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ! [D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
             => ( ( s @ ( fun @ A @ bool ) @ C )
                = ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1942]) ).

thf(256,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
        <=> ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ x1 ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_neg) ).

thf(4683,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
           => ? [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) )
          & ( ? [C: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_neg ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ x1 ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ real ) @ mku_real ) @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(1774,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_LT) ).

thf(14069,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1774]) ).

thf(2414,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ A @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEMPTYu_DELETE) ).

thf(17580,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ A @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2414]) ).

thf(1476,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GTu_conjunct8) ).

thf(12061,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1476]) ).

thf(1851,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_CMUL) ).

thf(14503,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ C ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1851]) ).

thf(2319,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ A @ B ) @ f ) ) ) @ ( s @ ( fun @ B @ A ) @ g ) ) ) )
     => ( ! [C: $i] :
            ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
          <=> ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ g ) @ ( s @ B @ D ) ) ) ) ) ) )
        & ! [C: $i] :
            ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
          <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ g ) @ ( s @ B @ D ) ) ) ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( s @ A @ C )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ g ) @ ( s @ B @ D ) ) ) )
          <=> ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) )
              = ( s @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aISOu_USAGE) ).

thf(17020,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ A ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ A @ B ) @ f ) ) ) @ ( s @ ( fun @ B @ A ) @ g ) ) ) )
     => ( ! [C: $i] :
            ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
             => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ g ) @ ( s @ B @ D ) ) ) ) ) ) )
            & ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ g ) @ ( s @ B @ D ) ) ) ) ) )
             => ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
        & ! [C: $i] :
            ( ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
             => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ g ) @ ( s @ B @ D ) ) ) ) ) ) )
            & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ g ) @ ( s @ B @ D ) ) ) ) ) )
             => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( ( s @ A @ C )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ g ) @ ( s @ B @ D ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) )
                = ( s @ B @ D ) ) )
            & ( ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) )
                = ( s @ B @ D ) )
             => ( ( s @ A @ C )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ g ) @ ( s @ B @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2319]) ).

thf(1243,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ODDu_conjunct2) ).

thf(10468,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1243]) ).

thf(1962,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_TRANS) ).

thf(15191,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1962]) ).

thf(2779,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ r_r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSPu_MATRIXu_CMUL) ).

thf(19581,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ r_r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2779]) ).

thf(365,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_ADD) ).

thf(5426,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).

thf(1901,axiom,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) )
           => ( ? [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
                    = ( s @ B @ C ) ) )
              & ! [D: $i,E: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
                      = ( s @ B @ C ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ E ) ) )
                      = ( s @ B @ C ) ) )
                 => ( ( s @ A @ D )
                    = ( s @ A @ E ) ) ) ) ) )
    <=> ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) ) ) )
           => ( ( s @ A @ C )
              = ( s @ A @ D ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) )
           => ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
                  = ( s @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIJECTIVEu_INJECTIVEu_SURJECTIVE) ).

thf(14742,plain,
    ! [A: $i,B: $i] :
      ( ( ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) )
             => ( ? [D: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
                      = ( s @ B @ C ) ) )
                & ! [D: $i,E: $i] :
                    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
                        = ( s @ B @ C ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ E ) ) )
                        = ( s @ B @ C ) ) )
                   => ( ( s @ A @ D )
                      = ( s @ A @ E ) ) ) ) ) )
       => ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) )
             => ? [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
                    = ( s @ B @ C ) ) ) ) ) )
      & ( ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) )
             => ? [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
                    = ( s @ B @ C ) ) ) ) )
       => ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ t0 ) ) ) )
             => ( ? [D: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
                      = ( s @ B @ C ) ) )
                & ! [D: $i,E: $i] :
                    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
                        = ( s @ B @ C ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ E ) ) )
                        = ( s @ B @ C ) ) )
                   => ( ( s @ A @ D )
                      = ( s @ A @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1901]) ).

thf(2804,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_EQ) ).

thf(19697,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ E ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2804]) ).

thf(264,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_SUBu_REFL) ).

thf(4715,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(2550,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_SYM) ).

thf(18349,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2550]) ).

thf(2408,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',altu_c) ).

thf(17565,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2408]) ).

thf(2624,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ measure0 ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEASURE) ).

thf(18765,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ measure0 ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ A @ D ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2624]) ).

thf(424,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_NUMSEGu_1) ).

thf(5813,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).

thf(1498,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_MUL2) ).

thf(12245,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1498]) ).

thf(3058,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ B )
       != ( s @ ( fun @ A @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFINITEu_NONEMPTY) ).

thf(21140,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ B )
       != ( s @ ( fun @ A @ bool ) @ empty ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3058]) ).

thf(2444,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LETu_ADD2) ).

thf(17775,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2444]) ).

thf(107,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_RID) ).

thf(3748,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(1798,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIN) ).

thf(14204,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1798]) ).

thf(1856,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLETu_CASES) ).

thf(14522,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1856]) ).

thf(1079,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_LMUL) ).

thf(9459,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1079]) ).

thf(3142,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) @ ( s @ ( cart @ real @ A ) @ z ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ z ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_ASSOC) ).

thf(21562,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) @ ( s @ ( cart @ real @ A ) @ z ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ z ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3142]) ).

thf(1352,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
    <=> ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aODDu_MOD) ).

thf(11259,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1352]) ).

thf(1717,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_LID) ).

thf(13625,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1717]) ).

thf(1236,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
        <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSIMPLEu_IMAGE) ).

thf(10411,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
           => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) ) )
          & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1236]) ).

thf(2540,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_ADDu_RDISTRIB) ).

thf(18305,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ B ) @ A ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2540]) ).

thf(2842,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_MAX) ).

thf(19916,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2842]) ).

thf(1390,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_OFu_NUMu_LE) ).

thf(11567,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1390]) ).

thf(2703,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ A @ G ) ) )
          = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ H ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ H ) ) ) ) ) )
         => ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                & ! [H: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_COMPOSEu_VSUM) ).

thf(19141,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ A @ G ) ) )
          = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i,H: $i] :
              ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ H ) ) )
              = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ ( cart @ real @ C ) ) ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ G ) ) ) @ ( s @ ( cart @ real @ B ) @ H ) ) ) ) ) )
         => ! [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                & ! [H: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) @ ( s @ A @ H ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ C ) ) ) @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2703]) ).

thf(745,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( list @ B ) @ num ) @ length ) @ ( s @ ( list @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLENGTHu_MAP) ).

thf(7638,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( list @ B ) @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( list @ B ) @ num ) @ length ) @ ( s @ ( list @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[745]) ).

thf(165,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ num @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( ( s @ num @ D )
             != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ num @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_NSUMu_MOD) ).

thf(4010,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ num @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( ( s @ num @ D )
             != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ num @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(276,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ s0 )
        = ( s @ ( fun @ A @ bool ) @ empty ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_CLAUSESu_conjunct0) ).

thf(4834,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ s0 )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ s0 )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(61,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_LE) ).

thf(3516,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(2588,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ! [E: $i] :
              ( ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) )
                = ( s @ A @ E ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) )
                = ( s @ A @ E ) ) ) )
     => ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aneutral) ).

thf(18564,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ! [E: $i] :
                ( ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) )
                  = ( s @ A @ E ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) )
                  = ( s @ A @ E ) ) ) )
          & ( ! [E: $i] :
                ( ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) )
                  = ( s @ A @ E ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) )
                  = ( s @ A @ E ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ A ) @ neutral ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2588]) ).

thf(2777,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ real @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_COMPOSEu_CMUL) ).

thf(19576,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ real @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2777]) ).

thf(936,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVISION) ).

thf(8781,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[936]) ).

thf(339,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ C )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
      <=> ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_ADDu_MOD) ).

thf(5243,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ C )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) )
        & ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).

thf(1189,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            <=> ? [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ H ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_COUNTABLEu_SUBSETS) ).

thf(10104,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
               => ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
              & ( ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1189]) ).

thf(2754,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( s @ ( fun @ B @ A ) @ C )
        = ( s @ ( fun @ B @ A ) @ D ) )
    <=> ! [E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFUNu_EQu_THM) ).

thf(19394,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ ( fun @ B @ A ) @ C )
          = ( s @ ( fun @ B @ A ) @ D ) )
       => ! [E: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) )
      & ( ! [E: $i] :
            ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) )
       => ( ( s @ ( fun @ B @ A ) @ C )
          = ( s @ ( fun @ B @ A ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2754]) ).

thf(656,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_1) ).

thf(7120,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[656]) ).

thf(2353,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_ELIMu_THMu_conjunct4) ).

thf(17185,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2353]) ).

thf(1129,axiom,
    ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ ( fun @ num @ bool ) @ univ ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFROMu_0) ).

thf(9824,plain,
    ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
    = ( s @ ( fun @ num @ bool ) @ univ ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1129]) ).

thf(654,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_SUC) ).

thf(7114,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[654]) ).

thf(2034,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ C @ B ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ C @ B ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMAGEu_o) ).

thf(15589,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ A ) ) ) @ o ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ C @ B ) @ E ) ) ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ C @ B ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2034]) ).

thf(2981,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ? [F: $i] :
              ( ( s @ ( prod @ A @ B ) @ D )
              = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ A ) @ fst ) @ ( s @ ( prod @ A @ B ) @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFSTu_DEF) ).

thf(20663,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ? [F: $i] :
                ( ( s @ ( prod @ A @ B ) @ D )
                = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) ) )
          & ( ? [F: $i] :
                ( ( s @ ( prod @ A @ B ) @ D )
                = ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ B @ F ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ A ) @ fst ) @ ( s @ ( prod @ A @ B ) @ D ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( prod @ A @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2981]) ).

thf(658,axiom,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
      <=> ( ( s @ ( list @ A ) @ t0 )
          = ( s @ ( list @ A ) @ nil ) ) )
      & ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ butlast ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
        = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ butlast ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBUTLASTu_conjunct1) ).

thf(7125,plain,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
       => ( ( s @ ( list @ A ) @ t0 )
          = ( s @ ( list @ A ) @ nil ) ) )
      & ( ( ( s @ ( list @ A ) @ t0 )
          = ( s @ ( list @ A ) @ nil ) )
       => ( p @ ( s @ bool @ B ) ) )
      & ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ butlast ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
        = ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ butlast ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[658]) ).

thf(578,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_MULu_EQu_conjunct1) ).

thf(6694,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).

thf(1931,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ E ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ C @ G ) ) )
          = ( s @ D @ ( i @ ( s @ ( fun @ B @ D ) @ gi_ ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ f ) @ ( s @ C @ G ) ) ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i] :
              ( ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ F ) @ ( s @ ( fun @ A @ B ) @ G ) ) )
              = ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ E ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) )
         => ! [G: $i] :
              ( ! [H: $i,I: $i] :
                  ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ A @ B ) ) @ G ) @ ( s @ ( fun @ C @ D ) @ H ) ) ) @ ( s @ A @ I ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ g ) @ ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ H ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ fi_ ) @ ( s @ A @ I ) ) ) ) ) ) ) )
             => ! [H: $i] :
                  ( ! [I: $i] :
                      ( ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ A @ B ) ) @ H ) @ ( s @ ( fun @ C @ D ) @ I ) ) )
                      = ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ A @ B ) ) @ G ) @ ( s @ ( fun @ C @ D ) @ I ) ) ) )
                 => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ A @ C ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ C @ A ) @ f ) ) ) @ ( s @ ( fun @ A @ C ) @ fi_ ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ B @ D ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ D @ B ) @ g ) ) ) @ ( s @ ( fun @ B @ D ) @ gi_ ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ A @ B ) ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ A @ B ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aISOu_FUN) ).

thf(15031,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ E ) @ ( s @ ( fun @ A @ B ) @ F ) ) ) @ ( s @ C @ G ) ) )
          = ( s @ D @ ( i @ ( s @ ( fun @ B @ D ) @ gi_ ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ F ) @ ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ f ) @ ( s @ C @ G ) ) ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i] :
              ( ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ F ) @ ( s @ ( fun @ A @ B ) @ G ) ) )
              = ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ E ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) )
         => ! [G: $i] :
              ( ! [H: $i,I: $i] :
                  ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ A @ B ) ) @ G ) @ ( s @ ( fun @ C @ D ) @ H ) ) ) @ ( s @ A @ I ) ) )
                  = ( s @ B @ ( i @ ( s @ ( fun @ D @ B ) @ g ) @ ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ H ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ fi_ ) @ ( s @ A @ I ) ) ) ) ) ) ) )
             => ! [H: $i] :
                  ( ! [I: $i] :
                      ( ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ A @ B ) ) @ H ) @ ( s @ ( fun @ C @ D ) @ I ) ) )
                      = ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ A @ B ) ) @ G ) @ ( s @ ( fun @ C @ D ) @ I ) ) ) )
                 => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ A @ C ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ C @ A ) @ f ) ) ) @ ( s @ ( fun @ A @ C ) @ fi_ ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ D ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ B ) @ ( fun @ ( fun @ B @ D ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ D @ B ) @ g ) ) ) @ ( s @ ( fun @ B @ D ) @ gi_ ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ A @ B ) ) @ ( fun @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ bool ) ) @ iso0 ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ A @ B ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ C @ D ) ) @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1931]) ).

thf(2481,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
            | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',achain) ).

thf(17951,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ! [D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
              | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) ) )
      & ( ! [D: $i,E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
              | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ chain ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2481]) ).

thf(1325,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
        ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
          <=> ( ( s @ B @ E )
              = ( s @ B @ D ) ) )
          & ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ ( cart @ real @ A ) ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ B @ E ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ f ) @ ( s @ B @ E ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
         => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ ( cart @ real @ A ) ) ) @ C ) @ ( s @ B @ E ) ) ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ f ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ f ) @ ( s @ B @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_CASESu_1) ).

thf(11013,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
        ? [F: $i] :
          ( ( ( p @ ( s @ bool @ F ) )
           => ( ( s @ B @ E )
              = ( s @ B @ D ) ) )
          & ( ( ( s @ B @ E )
              = ( s @ B @ D ) )
           => ( p @ ( s @ bool @ F ) ) )
          & ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ ( cart @ real @ A ) ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ B @ E ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ f ) @ ( s @ B @ E ) ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
         => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ ( cart @ real @ A ) ) ) @ C ) @ ( s @ B @ E ) ) ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ f ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ f ) @ ( s @ B @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1325]) ).

thf(237,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( ( s @ num @ B )
         != ( s @ num @ C ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) )
       != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBASISu_NE) ).

thf(4527,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( ( s @ num @ B )
         != ( s @ num @ C ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) )
       != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(2164,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) ) @ iterate ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUM) ).

thf(16288,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) ) @ iterate ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2164]) ).

thf(2967,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LETu_TRANS) ).

thf(20584,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2967]) ).

thf(2582,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_DIV) ).

thf(18501,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2582]) ).

thf(2913,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ c0 ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ c0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_ACu_conjunct2) ).

thf(20282,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ c0 ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ c0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2913]) ).

thf(1230,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_SGN) ).

thf(10386,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1230]) ).

thf(2879,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADJOINTu_LINEAR) ).

thf(20150,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ adjoint ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2879]) ).

thf(1212,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIT1u_THM) ).

thf(10268,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1212]) ).

thf(3006,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( prod @ B @ C ) @ D ) @ A ) @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ A ) ) @ passoc ) @ ( s @ ( fun @ ( prod @ ( prod @ B @ C ) @ D ) @ A ) @ E ) ) ) @ ( s @ ( prod @ B @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ G ) ) ) @ ( s @ D @ H ) ) ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ ( prod @ B @ C ) @ D ) @ A ) @ E ) @ ( s @ ( prod @ ( prod @ B @ C ) @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ ( prod @ B @ C ) @ D ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( fun @ D @ ( prod @ ( prod @ B @ C ) @ D ) ) ) @ c_ ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ) ) @ ( s @ D @ H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPASSOCu_DEF) ).

thf(20802,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ A ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( prod @ B @ C ) @ D ) @ A ) @ ( fun @ ( prod @ B @ ( prod @ C @ D ) ) @ A ) ) @ passoc ) @ ( s @ ( fun @ ( prod @ ( prod @ B @ C ) @ D ) @ A ) @ E ) ) ) @ ( s @ ( prod @ B @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( prod @ C @ D ) @ ( prod @ B @ ( prod @ C @ D ) ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ ( prod @ C @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ C @ D ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ ( prod @ C @ D ) ) ) @ c_ ) @ ( s @ C @ G ) ) ) @ ( s @ D @ H ) ) ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ ( prod @ B @ C ) @ D ) @ A ) @ E ) @ ( s @ ( prod @ ( prod @ B @ C ) @ D ) @ ( i @ ( s @ ( fun @ D @ ( prod @ ( prod @ B @ C ) @ D ) ) @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ ( fun @ D @ ( prod @ ( prod @ B @ C ) @ D ) ) ) @ c_ ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ F ) ) ) @ ( s @ C @ G ) ) ) ) ) @ ( s @ D @ H ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3006]) ).

thf(1445,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_COMBINEu_R) ).

thf(11918,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1445]) ).

thf(529,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_COMPONENT) ).

thf(6354,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).

thf(2496,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_NEG) ).

thf(18024,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2496]) ).

thf(1911,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ A @ C ) @ ( cart @ A @ ( finite_sum @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ ( cart @ A @ C ) @ ( cart @ A @ ( finite_sum @ B @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ B ) ) @ fstcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ) ) ) ) @ ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ C ) ) @ sndcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ) ) ) )
      = ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPASTECARTu_FSTu_SND) ).

thf(14845,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ A @ C ) @ ( cart @ A @ ( finite_sum @ B @ C ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ ( cart @ A @ C ) @ ( cart @ A @ ( finite_sum @ B @ C ) ) ) ) @ pastecart ) @ ( s @ ( cart @ A @ B ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ B ) ) @ fstcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ) ) ) ) @ ( s @ ( cart @ A @ C ) @ ( i @ ( s @ ( fun @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ ( cart @ A @ C ) ) @ sndcart ) @ ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ) ) ) )
      = ( s @ ( cart @ A @ ( finite_sum @ B @ C ) ) @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1911]) ).

thf(2756,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) ) ) @ ( s @ A @ B ) ) ) )
    <=> ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_PRED) ).

thf(19403,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) ) ) @ ( s @ A @ B ) ) ) )
       => ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ B ) ) ) ) ) )
      & ( ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) ) ) @ ( s @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2756]) ).

thf(2874,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_CMULu_ADDu_RDISTRIB) ).

thf(20123,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2874]) ).

thf(1377,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_1) ).

thf(11454,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1377]) ).

thf(1452,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POS) ).

thf(11946,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1452]) ).

thf(1059,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_neg) ).

thf(9350,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1059]) ).

thf(1259,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUBu_conjunct4) ).

thf(10563,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1259]) ).

thf(188,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_CLAUSESu_conjunct1) ).

thf(4182,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(428,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
        & ( ( s @ num @ B )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_EXP) ).

thf(5828,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
          & ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
          & ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).

thf(3041,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
    <=> ? [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_BETWEEN) ).

thf(21000,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
       => ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) )
      & ( ? [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3041]) ).

thf(456,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) ) @ fcons ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFCONSu_conjunct1) ).

thf(5963,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) ) @ fcons ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).

thf(511,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
          <=> ( ( s @ real @ B )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_GP) ).

thf(6257,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
           => ( ( s @ real @ B )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          & ( ( ( s @ real @ B )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ E ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).

thf(1641,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_OFu_NUMu_ADD) ).

thf(13193,plain,
    ! [A: $i,B: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1641]) ).

thf(1432,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_LZERO) ).

thf(11850,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1432]) ).

thf(1420,axiom,
    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ n ) ) ) ) ) )
  <=> ! [A: $i] :
        ( ( ( ( s @ num @ n )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) )
          | ( ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( ( s @ num @ n )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPREu_ELIMu_THM) ).

thf(11780,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ n ) ) ) ) ) )
     => ! [A: $i] :
          ( ( ( ( s @ num @ n )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) )
            | ( ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ n )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) )
    & ( ! [A: $i] :
          ( ( ( ( s @ num @ n )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) )
            | ( ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              & ( ( s @ num @ n )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ n ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1420]) ).

thf(1658,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_DIV) ).

thf(13288,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1658]) ).

thf(3156,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) )
      = ( s @ ( fun @ A @ bool ) @ p1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_ACIu_conjunct3) ).

thf(21618,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) )
      = ( s @ ( fun @ A @ bool ) @ p1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3156]) ).

thf(1721,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_MULTu_LMOD) ).

thf(13637,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1721]) ).

thf(3011,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ E ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALLu_IMP) ).

thf(20834,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ E ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3011]) ).

thf(982,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ max ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_OFu_NUMu_MAX) ).

thf(8997,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ max ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[982]) ).

thf(609,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
    <=> ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_EQu_0u_IFFu_NUMSEG) ).

thf(6854,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ! [D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ C ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ D ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ num ) @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).

thf(689,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_EQu_ABS) ).

thf(7266,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).

thf(2881,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_SUB) ).

thf(20163,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2881]) ).

thf(2143,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_EQu_REFL) ).

thf(16149,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2143]) ).

thf(2573,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ A ) )
        & ( ( p @ ( s @ bool @ B ) )
          | ( p @ ( s @ bool @ C ) ) ) )
    <=> ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ B ) ) )
        | ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_ORu_DISTRIB) ).

thf(18447,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( p @ ( s @ bool @ A ) )
          & ( ( p @ ( s @ bool @ B ) )
            | ( p @ ( s @ bool @ C ) ) ) )
       => ( ( ( p @ ( s @ bool @ A ) )
            & ( p @ ( s @ bool @ B ) ) )
          | ( ( p @ ( s @ bool @ A ) )
            & ( p @ ( s @ bool @ C ) ) ) ) )
      & ( ( ( ( p @ ( s @ bool @ A ) )
            & ( p @ ( s @ bool @ B ) ) )
          | ( ( p @ ( s @ bool @ A ) )
            & ( p @ ( s @ bool @ C ) ) ) )
       => ( ( p @ ( s @ bool @ A ) )
          & ( ( p @ ( s @ bool @ B ) )
            | ( p @ ( s @ bool @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2573]) ).

thf(2669,axiom,
    p @ ( s @ bool @ t ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRUTH) ).

thf(18960,plain,
    p @ ( s @ bool @ t ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2669]) ).

thf(2629,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ A @ B )
        = ( s @ A @ C ) )
     => ( ( s @ A @ C )
        = ( s @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEQu_SYM) ).

thf(18779,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ A @ B )
        = ( s @ A @ C ) )
     => ( ( s @ A @ C )
        = ( s @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2629]) ).

thf(58,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ADDu_conjunct5) ).

thf(3497,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(2935,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) )
            & ! [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ! [G: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) )
                     != ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_CLOSED) ).

thf(20425,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) ) )
            & ! [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ F ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ! [G: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) )
                     != ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ B ) @ neutral ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2935]) ).

thf(3171,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
        <=> ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
     => ! [E: $i] :
          ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ C ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ F ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_CURRY) ).

thf(21698,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
           => ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) )
          & ( ! [G: $i,H: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ geq ) @ ( s @ C @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) ) ) )
     => ! [E: $i] :
          ( ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ C ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ F ) ) ) ) ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( fun @ ( prod @ A @ B ) @ C ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ F ) ) ) ) ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ E ) @ ( s @ ( fun @ ( prod @ A @ B ) @ C ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3171]) ).

thf(518,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_ADD) ).

thf(6290,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).

thf(2714,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LTu_TOTAL) ).

thf(19196,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2714]) ).

thf(44,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_SUM) ).

thf(3430,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(1751,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_LE) ).

thf(13846,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1751]) ).

thf(2055,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_CONG) ).

thf(15667,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ G ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ G ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) @ ( s @ ( fun @ D @ bool ) @ H ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2055]) ).

thf(1071,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        <=> ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERSu_UNIONS) ).

thf(9393,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) )
          & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1071]) ).

thf(1910,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) ) ) )
        & ! [F: $i,G: $i,H: $i] :
            ( ! [I: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ I ) ) ) @ ( s @ B @ H ) ) ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) )
           => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ H ) ) )
                = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ H ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ H ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
        & ! [F: $i,G: $i,H: $i] :
            ( ! [I: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ I ) ) ) @ ( s @ B @ H ) ) ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ H ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) )
        & ! [F: $i,G: $i,H: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ? [F: $i] :
        ! [G: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_RECu_TAILu_GENERAL) ).

thf(14823,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) ) ) )
        & ! [F: $i,G: $i,H: $i] :
            ( ! [I: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ I ) ) ) @ ( s @ B @ H ) ) ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) )
           => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ H ) ) )
                = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) )
              & ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ H ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ H ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) ) )
        & ! [F: $i,G: $i,H: $i] :
            ( ! [I: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ I ) ) ) @ ( s @ B @ H ) ) ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ I ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ G ) @ ( s @ B @ I ) ) ) ) )
           => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ H ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ G ) ) ) @ ( s @ B @ H ) ) ) ) )
        & ! [F: $i,G: $i,H: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ? [F: $i] :
        ! [G: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ B ) ) @ D ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ E ) @ ( s @ ( fun @ B @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1910]) ).

thf(915,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ ( list @ A ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSeq2o_TL) ).

thf(8649,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ tl ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ ( list @ A ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[915]) ).

thf(465,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) ) @ assoc ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ C ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ A ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ ( prod @ A @ B ) @ A ) @ fst ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMEMu_ASSOC) ).

thf(6018,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ A @ B ) @ ( i @ ( s @ ( fun @ B @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( prod @ A @ B ) ) ) @ c_ ) @ ( s @ A @ D ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) ) @ assoc ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ C ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ A ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ ( prod @ A @ B ) @ A ) @ fst ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).

thf(1777,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SUPPORT) ).

thf(14094,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ real @ ( fun @ real @ real ) ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1777]) ).

thf(1287,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_RNEG) ).

thf(10738,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1287]) ).

thf(311,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_DIV2u_EQ) ).

thf(5040,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).

thf(1027,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_ADDu_RCANCEL) ).

thf(9223,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1027]) ).

thf(140,axiom,
    ! [A: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
        = ( s @ num @ u_0 ) )
    <=> ( p @ ( s @ bool @ f0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EQu_conjunct3) ).

thf(3896,plain,
    ! [A: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ u_0 ) )
       => ( p @ ( s @ bool @ f0 ) ) )
      & ( ( p @ ( s @ bool @ f0 ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(3123,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) )
      <=> ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_FINITEu_UNIONS) ).

thf(21481,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) )
         => ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
        & ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3123]) ).

thf(2517,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_LE) ).

thf(18147,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2517]) ).

thf(664,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_RESTRICT) ).

thf(7157,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[664]) ).

thf(516,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ D ) @ ( s @ B @ F ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ E ) @ ( s @ B @ F ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ real ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_NORMu_LE) ).

thf(6277,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ D ) @ ( s @ B @ F ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ B @ real ) @ E ) @ ( s @ B @ F ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ B @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ real ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).

thf(3157,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_ASSOC) ).

thf(21621,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3157]) ).

thf(1155,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_LE0) ).

thf(9955,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1155]) ).

thf(1558,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ E ) @ ( s @ D @ F ) ) )
          = ( s @ bool @ t ) )
     => ! [F: $i,G: $i,H: $i] :
          ( ? [I: $i,J: $i,K: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ D ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ B @ ( fun @ A @ D ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ D ) ) @ J ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ F ) ) ) )
              & ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ I ) @ ( s @ A @ K ) ) )
                = ( s @ C @ H ) )
              & ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) )
                = ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ D ) ) @ J ) @ ( s @ B @ G ) ) ) @ ( s @ A @ K ) ) ) ) )
          | ( ~ ? [I: $i,J: $i,K: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ D ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ B @ ( fun @ A @ D ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ D ) ) @ J ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ F ) ) ) )
                  & ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ I ) @ ( s @ A @ K ) ) )
                    = ( s @ C @ H ) ) )
            & ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) )
              = ( s @ D @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ D ) @ h_ ) @ ( s @ ( fun @ D @ bool ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCASEWISEu_CASES) ).

thf(12620,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ E ) @ ( s @ D @ F ) ) )
          = ( s @ bool @ t ) )
     => ! [F: $i,G: $i,H: $i] :
          ( ? [I: $i,J: $i,K: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ D ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ B @ ( fun @ A @ D ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ D ) ) @ J ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ F ) ) ) )
              & ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ I ) @ ( s @ A @ K ) ) )
                = ( s @ C @ H ) )
              & ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) )
                = ( s @ D @ ( i @ ( s @ ( fun @ A @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ D ) ) @ J ) @ ( s @ B @ G ) ) ) @ ( s @ A @ K ) ) ) ) )
          | ( ~ ? [I: $i,J: $i,K: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ bool ) ) @ mem ) @ ( s @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ D ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ B @ ( fun @ A @ D ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ C ) @ I ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ D ) ) @ J ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ F ) ) ) )
                  & ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ I ) @ ( s @ A @ K ) ) )
                    = ( s @ C @ H ) ) )
            & ( ( s @ D @ ( i @ ( s @ ( fun @ C @ D ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ D ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ ( fun @ B @ ( fun @ C @ D ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ B @ ( fun @ A @ D ) ) ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) @ ( s @ C @ H ) ) )
              = ( s @ D @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ D ) @ h_ ) @ ( s @ ( fun @ D @ bool ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1558]) ).

thf(385,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_conjunct1) ).

thf(5549,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).

thf(392,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
            <=> ? [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ H ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ F ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_RESTRICTu_SET) ).

thf(5568,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
               => ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
              & ( ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ E ) @ ( s @ A @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ F ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).

thf(2546,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ? [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPu_FINITEu_LEMMA) ).

thf(18337,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ? [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2546]) ).

thf(3073,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ E ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_NEGu_COMPONENT) ).

thf(21217,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ matrixu_neg ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ E ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3073]) ).

thf(787,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUBu_conjunct3) ).

thf(7887,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[787]) ).

thf(2233,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_POW) ).

thf(16646,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2233]) ).

thf(2344,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_LADD) ).

thf(17159,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2344]) ).

thf(118,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ n10 @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ n10 @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_1) ).

thf(3793,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ n10 @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ n10 @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(3051,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_ADDu_RCANCEL) ).

thf(21093,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3051]) ).

thf(2312,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
    <=> ? [C: $i] :
          ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPSUBSETu_UNIV) ).

thf(16993,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) )
       => ? [C: $i] :
            ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
      & ( ? [C: $i] :
            ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2312]) ).

thf(561,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ ( prod @ B @ A ) @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ ( sum0 @ A @ C ) ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ ( prod @ B @ A ) @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ B @ ( sum0 @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ C ) @ bool ) @ ( fun @ ( prod @ B @ ( sum0 @ A @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ C ) @ bool ) @ ( fun @ ( prod @ B @ ( sum0 @ A @ C ) ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ A @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ A @ C ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ ( prod @ B @ A ) @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ ( prod @ B @ A ) @ ( prod @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ ( prod @ B @ A ) @ ( prod @ B @ C ) ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ B @ A ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LDISTRIB) ).

thf(6536,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ ( prod @ B @ A ) @ ( prod @ B @ C ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ ( sum0 @ A @ C ) ) @ bool ) @ ( fun @ ( fun @ ( sum0 @ ( prod @ B @ A ) @ ( prod @ B @ C ) ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( prod @ B @ ( sum0 @ A @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ C ) @ bool ) @ ( fun @ ( prod @ B @ ( sum0 @ A @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ C ) @ bool ) @ ( fun @ ( prod @ B @ ( sum0 @ A @ C ) ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ A @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( sum0 @ A @ C ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ ( prod @ B @ A ) @ ( prod @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ ( prod @ B @ A ) @ ( prod @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( fun @ ( sum0 @ ( prod @ B @ A ) @ ( prod @ B @ C ) ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ ( prod @ B @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ B @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ B @ A ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ ( prod @ B @ C ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ ( prod @ B @ C ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ C @ bool ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).

thf(2261,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_IMAGE) ).

thf(16778,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2261]) ).

thf(2373,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MAXu_ACIu_conjunct4) ).

thf(17291,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ x ) ) ) @ ( s @ real @ y ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2373]) ).

thf(799,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_ACIu_conjunct2) ).

thf(7939,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ q ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ p1 ) ) ) @ ( s @ ( fun @ A @ bool ) @ r0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[799]) ).

thf(1122,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_RNEG) ).

thf(9799,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1122]) ).

thf(2395,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ? [E: $i] :
          ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
          & ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
             => ( ? [G: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) )
                      = ( s @ A @ F ) ) )
                & ! [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) )
                        = ( s @ A @ F ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                        = ( s @ A @ F ) ) )
                   => ( ( s @ B @ G )
                      = ( s @ B @ H ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aequ_c) ).

thf(17447,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ? [E: $i] :
            ( ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
            & ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
               => ( ? [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) )
                        = ( s @ A @ F ) ) )
                  & ! [G: $i,H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                        & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) )
                          = ( s @ A @ F ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                        & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                          = ( s @ A @ F ) ) )
                     => ( ( s @ B @ G )
                        = ( s @ B @ H ) ) ) ) ) ) )
      & ( ? [E: $i] :
            ( ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
            & ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
               => ( ? [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) )
                        = ( s @ A @ F ) ) )
                  & ! [G: $i,H: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                        & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) )
                          = ( s @ A @ F ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                        & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ H ) ) )
                          = ( s @ A @ F ) ) )
                     => ( ( s @ B @ G )
                        = ( s @ B @ H ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2395]) ).

thf(1744,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ ( fun @ num @ bool ) @ empty ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_EMPTY) ).

thf(13816,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ ( fun @ num @ bool ) @ empty ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) )
       => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ ( fun @ num @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1744]) ).

thf(2110,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_LMUL) ).

thf(15977,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ C ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2110]) ).

thf(322,axiom,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ bool @ f0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_LTu_conjunct1) ).

thf(5119,plain,
    ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ bool @ f0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).

thf(330,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_RMUL) ).

thf(5211,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).

thf(178,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
            <=> ! [F: $i,G: $i] :
                ? [H: $i] :
                  ( ( ( p @ ( s @ bool @ H ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                      | ( ( ( s @ A @ G )
                          = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) )
                        & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ A @ F ) ) ) )
                          | ( ( s @ A @ F )
                            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
         => ! [D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) )
                & ? [E: $i] :
                    ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORDINALu_SUC) ).

thf(4074,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
          & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) )
               => ! [F: $i,G: $i] :
                  ? [H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                        | ( ( ( s @ A @ G )
                            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) )
                          & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ A @ F ) ) ) )
                            | ( ( s @ A @ F )
                              = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                        | ( ( ( s @ A @ G )
                            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) )
                          & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ A @ F ) ) ) )
                            | ( ( s @ A @ F )
                              = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) ) )
              & ( ! [F: $i,G: $i] :
                  ? [H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                        | ( ( ( s @ A @ G )
                            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) )
                          & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ A @ F ) ) ) )
                            | ( ( s @ A @ F )
                              = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
                        | ( ( ( s @ A @ G )
                            = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) )
                          & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ A @ F ) ) ) )
                            | ( ( s @ A @ F )
                              = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) @ ( s @ bool @ H ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ E ) ) ) ) ) )
         => ! [D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) )
                & ? [E: $i] :
                    ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ D ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(1333,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_conjunct0) ).

thf(11071,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1333]) ).

thf(1811,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_COMPOSEu_ADD) ).

thf(14261,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ F ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) ) ) @ C ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1811]) ).

thf(2192,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_SUC) ).

thf(16453,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2192]) ).

thf(2574,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ( p @ ( s @ bool @ C ) ) )
    <=> ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        | ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEFTu_FORALLu_ORu_THM) ).

thf(18464,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            | ( p @ ( s @ bool @ C ) ) )
       => ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ( p @ ( s @ bool @ C ) ) ) )
      & ( ( ! [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
          | ( p @ ( s @ bool @ C ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
            | ( p @ ( s @ bool @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2574]) ).

thf(3145,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_CLAUSESu_conjunct3) ).

thf(21574,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3145]) ).

thf(1368,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ B ) ) ) )
        <=> ? [C: $i,D: $i] :
              ( ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) )
                = ( s @ bool @ B ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ y ) ) ) @ ( s @ nadd @ D ) ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ x ) ) ) @ ( s @ hreal @ y ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ bool ) @ bool ) @ h_ ) @ ( s @ ( fun @ bool @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ahrealu_le) ).

thf(11381,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ B ) ) ) )
           => ? [C: $i,D: $i] :
                ( ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) )
                  = ( s @ bool @ B ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ y ) ) ) @ ( s @ nadd @ D ) ) ) ) ) )
          & ( ? [C: $i,D: $i] :
                ( ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) )
                  = ( s @ bool @ B ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ x ) ) ) @ ( s @ nadd @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ nadd @ bool ) ) @ destu_hreal ) @ ( s @ hreal @ y ) ) ) @ ( s @ nadd @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ B ) ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ x ) ) ) @ ( s @ hreal @ y ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ bool ) @ bool ) @ h_ ) @ ( s @ ( fun @ bool @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1368]) ).

thf(1011,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_MULTu_conjunct1) ).

thf(9162,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1011]) ).

thf(2329,axiom,
    ( ? [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ A ) ) ) )
  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ t ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ f0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_BOOLu_THM) ).

thf(17084,plain,
    ( ( ? [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ A ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ t ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ f0 ) ) ) ) ) )
    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ t ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ f0 ) ) ) ) )
     => ? [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ p0 ) @ ( s @ bool @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2329]) ).

thf(1679,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
          = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_OFFSETu_IMAGE) ).

thf(13389,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) )
          = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1679]) ).

thf(542,axiom,
    ! [A: $i] :
      ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ B ) ) ) )
    <=> ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n10 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_VECTORu_1) ).

thf(6462,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ B ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n10 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) )
      & ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n10 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).

thf(289,axiom,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ m ) ) ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ m ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ahrealu_ofu_numu_th) ).

thf(4927,plain,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ ( fun @ nadd @ bool ) @ hreal ) @ mku_hreal ) @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ m ) ) ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(662,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_RESTRICT) ).

thf(7152,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[662]) ).

thf(1489,axiom,
    ( ( ( ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ q ) ) )
     => ( p @ ( s @ bool @ r0 ) ) )
  <=> ( ( p @ ( s @ bool @ p1 ) )
     => ( ( p @ ( s @ bool @ q ) )
       => ( p @ ( s @ bool @ r0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aIMPu_CONJ) ).

thf(12125,plain,
    ( ( ( ( ( p @ ( s @ bool @ p1 ) )
          & ( p @ ( s @ bool @ q ) ) )
       => ( p @ ( s @ bool @ r0 ) ) )
     => ( ( p @ ( s @ bool @ p1 ) )
       => ( ( p @ ( s @ bool @ q ) )
         => ( p @ ( s @ bool @ r0 ) ) ) ) )
    & ( ( ( p @ ( s @ bool @ p1 ) )
       => ( ( p @ ( s @ bool @ q ) )
         => ( p @ ( s @ bool @ r0 ) ) ) )
     => ( ( ( p @ ( s @ bool @ p1 ) )
          & ( p @ ( s @ bool @ q ) ) )
       => ( p @ ( s @ bool @ r0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1489]) ).

thf(245,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ B ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ q0 ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMONOu_ALL) ).

thf(4619,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ B ) ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ q0 ) ) ) @ ( s @ ( list @ A ) @ l0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(500,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_2) ).

thf(6198,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ insert ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ empty ) ) ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).

thf(1051,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ num @ C ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ B ) @ ( s @ num @ C ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_CONST) ).

thf(9321,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ num @ C ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ num ) ) @ B ) @ ( s @ num @ C ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1051]) ).

thf(2974,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_ADDu_ABSORBu_LE) ).

thf(20612,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2974]) ).

thf(2383,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
    <=> ? [C: $i] :
          ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_EXISTS) ).

thf(17348,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
       => ? [C: $i] :
            ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
      & ( ? [C: $i] :
            ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2383]) ).

thf(431,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_LE) ).

thf(5864,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).

thf(2793,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_MIN) ).

thf(19637,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2793]) ).

thf(1905,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ x ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_SUBu_ADD2) ).

thf(14806,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ x ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1905]) ).

thf(1094,axiom,
    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ x ) ) ) @ ( s @ num @ y ) ) ) ) ) ) ) )
  <=> ! [A: $i] :
        ( ( ( ( s @ num @ x )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ y ) ) ) @ ( s @ num @ A ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
        & ( ( ( s @ num @ y )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ x ) ) ) @ ( s @ num @ A ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_ELIMu_THM) ).

thf(9542,plain,
    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ x ) ) ) @ ( s @ num @ y ) ) ) ) ) ) ) )
     => ! [A: $i] :
          ( ( ( ( s @ num @ x )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ y ) ) ) @ ( s @ num @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
          & ( ( ( s @ num @ y )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ x ) ) ) @ ( s @ num @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) ) )
    & ( ! [A: $i] :
          ( ( ( ( s @ num @ x )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ y ) ) ) @ ( s @ num @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) )
          & ( ( ( s @ num @ y )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ x ) ) ) @ ( s @ num @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ A ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ p0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ x ) ) ) @ ( s @ num @ y ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1094]) ).

thf(2767,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ B )
       != ( s @ ( fun @ A @ bool ) @ C ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_EQUALu_SETS) ).

thf(19505,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ C ) )
       => ? [D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
             => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
            & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
             => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
            & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ B )
         != ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2767]) ).

thf(1125,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_SIGN) ).

thf(9812,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1125]) ).

thf(1031,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_LE) ).

thf(9239,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1031]) ).

thf(1483,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_ge ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_GE) ).

thf(12108,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_ge ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1483]) ).

thf(2280,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_TRANS) ).

thf(16871,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2280]) ).

thf(1760,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ D @ H ) ) ) )
        <=> ? [I: $i,J: $i] :
              ( ( ( p @ ( s @ bool @ J ) )
              <=> ? [K: $i,L: $i,M: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ in ) @ ( s @ D @ I ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ bool @ ( fun @ D @ bool ) ) ) @ setspec ) @ ( s @ D @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ D @ I ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) )
            <=> ? [J: $i,K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) )
         => ! [G: $i,H: $i] :
              ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( fun @ D @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) ) ) ) )
              = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ D @ bool ) ) @ gspec ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_GSPECu_conjunct2) ).

thf(13890,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ D @ H ) ) ) )
           => ? [I: $i,J: $i] :
                ( ( ( p @ ( s @ bool @ J ) )
                 => ? [K: $i,L: $i,M: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ in ) @ ( s @ D @ I ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) ) ) )
                & ( ? [K: $i,L: $i,M: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ in ) @ ( s @ D @ I ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ J ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ bool @ ( fun @ D @ bool ) ) ) @ setspec ) @ ( s @ D @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ D @ I ) ) ) ) ) )
          & ( ? [I: $i,J: $i] :
                ( ( ( p @ ( s @ bool @ J ) )
                 => ? [K: $i,L: $i,M: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ in ) @ ( s @ D @ I ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) ) ) )
                & ( ? [K: $i,L: $i,M: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ in ) @ ( s @ D @ I ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ J ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ bool @ ( fun @ D @ bool ) ) ) @ setspec ) @ ( s @ D @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ D @ I ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ D @ H ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) )
               => ? [J: $i,K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) )
              & ( ? [J: $i,K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) ) ) )
         => ! [G: $i,H: $i] :
              ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( fun @ D @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) ) ) ) )
              = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ D @ bool ) ) @ gspec ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1760]) ).

thf(1923,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_INSERTu_DELETE) ).

thf(14984,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1923]) ).

thf(1413,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ B @ b0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ B @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ n ) ) ) ) ) )
    <=> ? [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ B @ b0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ num @ n ) ) ) )
          & ( ( s @ B @ a0 )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINRECu_conjunct1) ).

thf(11744,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ B @ b0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ B @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ n ) ) ) ) ) )
       => ? [C: $i,D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ B @ b0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ num @ n ) ) ) )
            & ( ( s @ B @ a0 )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) ) )
      & ( ? [C: $i,D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ B @ b0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ B @ D ) ) ) @ ( s @ num @ n ) ) ) )
            & ( ( s @ B @ a0 )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) @ ( s @ A @ C ) ) ) @ ( s @ B @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ B @ b0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ B @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ n ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1413]) ).

thf(2551,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MINu_SYM) ).

thf(18352,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2551]) ).

thf(652,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_LINV) ).

thf(7081,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[652]) ).

thf(1513,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
        = ( s @ real @ a0 ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_EQu_SQUARE) ).

thf(12293,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ a0 ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ a0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1513]) ).

thf(3167,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ! [E: $i] :
                ( ! [F: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) ) ) ) )
            & ! [E: $i] :
                ( ! [F: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) )
         => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_Pu_ANDu_Q) ).

thf(21670,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ! [E: $i] :
                ( ! [F: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) ) ) ) )
            & ! [E: $i] :
                ( ! [F: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) )
         => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3167]) ).

thf(862,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ x ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_powu_conjunct1) ).

thf(8324,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ x ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ x ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[862]) ).

thf(1683,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
        <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ A @ G ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ A @ G ) ) ) )
            <=> ? [H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ I ) )
                  <=> ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) )
                      = ( s @ B @ F ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                & ! [G: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_FINITEu_PREIMAGE) ).

thf(13413,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
           => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ A @ G ) ) ) ) )
          & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ A @ G ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ A @ G ) ) ) )
               => ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) )
                        = ( s @ B @ F ) ) )
                    & ( ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) )
                        = ( s @ B @ F ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
              & ( ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) )
                        = ( s @ B @ F ) ) )
                    & ( ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ H ) ) )
                        = ( s @ B @ F ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                & ! [G: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1683]) ).

thf(2649,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_SUB) ).

thf(18855,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2649]) ).

thf(2166,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ A )
       != ( s @ int @ B ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NOTu_EQ) ).

thf(16294,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ A )
         != ( s @ int @ B ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) )
       => ( ( s @ int @ A )
         != ( s @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2166]) ).

thf(412,axiom,
    ! [A: $i] :
    ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ARCHu_POW2) ).

thf(5728,plain,
    ! [A: $i] :
    ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).

thf(2325,axiom,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ empty )
     != ( s @ ( fun @ A @ bool ) @ univ ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEMPTYu_NOTu_UNIV) ).

thf(17052,plain,
    ! [A: $i] :
      ( ( s @ ( fun @ A @ bool ) @ empty )
     != ( s @ ( fun @ A @ bool ) @ univ ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2325]) ).

thf(2366,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LTEu_TOTAL) ).

thf(17265,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2366]) ).

thf(1986,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPSUBSETu_ALT) ).

thf(15337,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1986]) ).

thf(2877,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_conjunct1) ).

thf(20138,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ B ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ B ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2877]) ).

thf(501,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
    <=> ( ( s @ ( list @ A ) @ B )
        = ( s @ ( list @ A ) @ nil ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLENGTHu_EQu_NIL) ).

thf(6201,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( ( s @ ( list @ A ) @ B )
          = ( s @ ( list @ A ) @ nil ) ) )
      & ( ( ( s @ ( list @ A ) @ B )
          = ( s @ ( list @ A ) @ nil ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).

thf(132,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_RMULu_EQ) ).

thf(3860,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(1573,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_SQUARE) ).

thf(12702,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1573]) ).

thf(1556,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ D ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_0) ).

thf(12615,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) @ ( s @ A @ D ) ) )
          = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1556]) ).

thf(744,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_MULTu_MOD2) ).

thf(7634,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[744]) ).

thf(1455,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LTu_CARD) ).

thf(11960,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1455]) ).

thf(17,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
      <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ B ) @ B ) @ hd ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ tl ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) ) ) )
      & ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
          <=> ( ( s @ ( list @ B ) @ l2 )
              = ( s @ ( list @ B ) @ nil ) ) )
          & ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ f0 ) ) ) @ ( s @ bool @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALL2u_DEFu_conjunct1) ).

thf(3232,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ B ) @ B ) @ hd ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ tl ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) @ ( s @ A @ h1 ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ B ) @ B ) @ hd ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) @ ( s @ ( list @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ B ) ) @ tl ) @ ( s @ ( list @ B ) @ l2 ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ C ) ) )
      & ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
           => ( ( s @ ( list @ B ) @ l2 )
              = ( s @ ( list @ B ) @ nil ) ) )
          & ( ( ( s @ ( list @ B ) @ l2 )
              = ( s @ ( list @ B ) @ nil ) )
           => ( p @ ( s @ bool @ D ) ) )
          & ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h1 ) ) ) @ ( s @ ( list @ A ) @ t1 ) ) ) ) ) @ ( s @ ( list @ B ) @ l2 ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ f0 ) ) ) @ ( s @ bool @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(1029,axiom,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( p @ ( s @ bool @ D ) )
      <=> ( ( s @ ( list @ A ) @ B )
          = ( s @ ( list @ A ) @ nil ) ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHDu_APPEND) ).

thf(9229,plain,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( p @ ( s @ bool @ D ) )
       => ( ( s @ ( list @ A ) @ B )
          = ( s @ ( list @ A ) @ nil ) ) )
      & ( ( ( s @ ( list @ A ) @ B )
          = ( s @ ( list @ A ) @ nil ) )
       => ( p @ ( s @ bool @ D ) ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1029]) ).

thf(26,axiom,
    ( ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ a0 ) ) ) ) )
    = ( s @ nadd @ a0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anaddu_abs) ).

thf(3291,plain,
    ( ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ a0 ) ) ) ) )
    = ( s @ nadd @ a0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(1196,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_ADDu_LEu_MULu_INFINITE) ).

thf(10173,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1196]) ).

thf(660,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ n ) ) ) ) ) )
    <=> ? [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ n ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( ( s @ ( fun @ A @ bool ) @ s0 )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_CLAUSESu_conjunct1) ).

thf(7140,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ n ) ) ) ) ) )
       => ? [B: $i,C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ n ) ) ) )
            & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ( ( s @ ( fun @ A @ bool ) @ s0 )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
      & ( ? [B: $i,C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ n ) ) ) )
            & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
            & ( ( s @ ( fun @ A @ bool ) @ s0 )
              = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ n ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[660]) ).

thf(892,axiom,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_INVu_0) ).

thf(8478,plain,
    ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ hrealu_inv ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ hreal @ ( i @ ( s @ ( fun @ num @ hreal ) @ hrealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).

thf(2194,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ n ) ) ) @ ( s @ ( cart @ real @ A ) @ p1 ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ n ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) @ ( s @ ( cart @ real @ A ) @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_ACu_conjunct2) ).

thf(16458,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ n ) ) ) @ ( s @ ( cart @ real @ A ) @ p1 ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ n ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ m ) ) ) @ ( s @ ( cart @ real @ A ) @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2194]) ).

thf(1102,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ) ) )
    <=> ( ( s @ real @ A )
       != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_SQUARE) ).

thf(9579,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ) ) )
       => ( ( s @ real @ A )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ real @ A )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1102]) ).

thf(75,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_MUL) ).

thf(3601,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(1119,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
            & ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
               => ( ( s @ A @ D )
                  = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aordinal) ).

thf(9767,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
          & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ less ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( prod @ A @ A ) @ ( i @ ( s @ ( fun @ A @ ( prod @ A @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ ( prod @ A @ A ) ) ) @ c_ ) @ ( s @ A @ E ) ) ) @ ( s @ A @ D ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
                 => ( ( s @ A @ D )
                    = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) ) )
          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
                 => ( ( s @ A @ D )
                    = ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ordinal ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1119]) ).

thf(1403,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
      <=> ( ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C )
            = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) )
          | ? [D: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) )
              & ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C )
                = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSEGu_LINSEG) ).

thf(11680,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ woset ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
     => ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) )
         => ( ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C )
              = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) )
            | ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) )
                & ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C )
                  = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) ) )
        & ( ( ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C )
              = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) )
            | ? [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ bool ) ) @ fl ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) )
                & ( ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C )
                  = ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ A @ ( fun @ ( prod @ A @ A ) @ bool ) ) ) @ linseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ ( fun @ ( fun @ ( prod @ A @ A ) @ bool ) @ bool ) ) @ inseg ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( prod @ A @ A ) @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1403]) ).

thf(1521,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
            <=> ? [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ H ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ E ) @ ( s @ A @ G ) ) )
                       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) ) ) ) ) ) ) )
             => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_ADDu_GEN) ).

thf(12374,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) ) @ ( s @ A @ F ) ) ) )
               => ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ E ) @ ( s @ A @ G ) ) )
                         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ E ) @ ( s @ A @ G ) ) )
                         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) ) )
              & ( ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ E ) @ ( s @ A @ G ) ) )
                         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                        & ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ E ) @ ( s @ A @ G ) ) )
                         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ A @ G ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) )
         => ! [D: $i,E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) ) ) ) ) ) ) )
             => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1521]) ).

thf(785,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_POW2) ).

thf(7883,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[785]) ).

thf(1412,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFu_FINITE) ).

thf(11739,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) ) )
     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
        & ! [B: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1412]) ).

thf(1050,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_NEGu_MINUS1) ).

thf(9318,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1050]) ).

thf(138,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( list @ A ) @ B )
       != ( s @ ( list @ A ) @ nil ) )
     => ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ B ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLASTu_EL) ).

thf(3892,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( list @ A ) @ B )
       != ( s @ ( list @ A ) @ nil ) )
     => ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ last0 ) @ ( s @ ( list @ A ) @ B ) ) )
        = ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ A ) @ A ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( list @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(485,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
        | ( ( s @ num @ C )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_MONOu_EQ) ).

thf(6131,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ B ) )
          | ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ B ) )
          | ( ( s @ num @ C )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).

thf(1649,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ ( cart @ real @ n10 ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ n10 ) @ x ) ) ) @ ( s @ ( cart @ real @ n10 ) @ y ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_1) ).

thf(13257,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ ( cart @ real @ n10 ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ n10 ) @ x ) ) ) @ ( s @ ( cart @ real @ n10 ) @ y ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ x ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ n10 ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ n10 ) @ y ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1649]) ).

thf(1173,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_SUBR) ).

thf(10041,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1173]) ).

thf(387,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LE2) ).

thf(5554,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).

thf(2146,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_LMUL) ).

thf(16160,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2146]) ).

thf(453,axiom,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_DIST) ).

thf(5954,plain,
    ! [A: $i] :
    ? [B: $i] :
    ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).

thf(2394,axiom,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        & ! [C: $i,D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
           => ( ( s @ A @ C )
              = ( s @ A @ D ) ) ) )
    <=> ? [C: $i] :
        ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        <=> ( ( s @ A @ C )
            = ( s @ A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_UNIQUEu_ALT) ).

thf(17414,plain,
    ! [A: $i,B: $i] :
      ( ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) ) )
       => ? [C: $i] :
          ! [D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) )
            & ( ( ( s @ A @ C )
                = ( s @ A @ D ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) )
      & ( ? [C: $i] :
          ! [D: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) )
            & ( ( ( s @ A @ C )
                = ( s @ A @ D ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) )
       => ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
             => ( ( s @ A @ C )
                = ( s @ A @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2394]) ).

thf(683,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ m ) ) ) @ ( s @ int @ n ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ n ) ) ) @ ( s @ int @ m ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ADDu_ACu_conjunct0) ).

thf(7242,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ m ) ) ) @ ( s @ int @ n ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ n ) ) ) @ ( s @ int @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[683]) ).

thf(1437,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ y ) ) ) @ ( s @ int @ x ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MAXu_ACIu_conjunct0) ).

thf(11871,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ y ) ) ) @ ( s @ int @ x ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1437]) ).

thf(2895,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_BOUNDSu_LE) ).

thf(20219,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2895]) ).

thf(2685,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ rest ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ choice0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREST) ).

thf(19047,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ rest ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ choice0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2685]) ).

thf(2760,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ empty ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MINu_INF) ).

thf(19432,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ B ) ) ) @ ( s @ ( fun @ real @ bool ) @ empty ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2760]) ).

thf(1952,axiom,
    ! [A: $i] :
      ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) )
    <=> ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXISTSu_DIFF) ).

thf(15108,plain,
    ! [A: $i] :
      ( ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
      & ( ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ? [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1952]) ).

thf(1409,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_INCLu_EXCL) ).

thf(11722,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1409]) ).

thf(1486,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POS) ).

thf(12117,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1486]) ).

thf(3081,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ! [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ E ) ) ) )
                 => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ F ) ) )
                    = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ F ) ) ) ) )
             => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ A @ E ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
         => ? [C: $i] :
            ! [D: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_RECu_WF) ).

thf(21241,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i] :
              ( ! [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ F ) ) ) @ ( s @ A @ E ) ) ) )
                 => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ F ) ) )
                    = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ F ) ) ) ) )
             => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ A @ E ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) )
         => ? [C: $i] :
            ! [D: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ D ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3081]) ).

thf(1711,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_GPu_BASIC) ).

thf(13577,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ A ) @ ( s @ real @ B ) ) ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1711]) ).

thf(2384,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NOTu_LT) ).

thf(17355,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2384]) ).

thf(643,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLENGTHu_APPEND) ).

thf(7029,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ append ) @ ( s @ ( list @ A ) @ B ) ) ) @ ( s @ ( list @ A ) @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[643]) ).

thf(540,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_MULTu_conjunct4) ).

thf(6456,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ u_0 ) ) )
      = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).

thf(1961,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_LSUM) ).

thf(15187,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ D ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ A @ real ) ) ) @ C ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1961]) ).

thf(277,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) )
    <=> ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
          <=> ( ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
              | ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ) )
          & ? [E: $i] :
              ( ( ( p @ ( s @ bool @ E ) )
              <=> ( ( s @ real @ B )
                  = ( s @ real @ C ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_EQu_EQ) ).

thf(4841,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) )
       => ? [D: $i] :
            ( ( ( p @ ( s @ bool @ D ) )
             => ( ( ( s @ num @ A )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                | ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ) )
            & ( ( ( ( s @ num @ A )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                | ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) )
             => ( p @ ( s @ bool @ D ) ) )
            & ? [E: $i] :
                ( ( ( p @ ( s @ bool @ E ) )
                 => ( ( s @ real @ B )
                    = ( s @ real @ C ) ) )
                & ( ( ( s @ real @ B )
                    = ( s @ real @ C ) )
                 => ( p @ ( s @ bool @ E ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ E ) ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( ( p @ ( s @ bool @ D ) )
             => ( ( ( s @ num @ A )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                | ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ) )
            & ( ( ( ( s @ num @ A )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                | ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) )
             => ( p @ ( s @ bool @ D ) ) )
            & ? [E: $i] :
                ( ( ( p @ ( s @ bool @ E ) )
                 => ( ( s @ real @ B )
                    = ( s @ real @ C ) ) )
                & ( ( ( s @ real @ B )
                    = ( s @ real @ C ) )
                 => ( p @ ( s @ bool @ E ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ E ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(2008,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLSu_EQ) ).

thf(15480,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ! [F: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ E ) ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) )
     => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2008]) ).

thf(1431,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) )
    <=> ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_NZ) ).

thf(11843,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) )
       => ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1431]) ).

thf(1733,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_OFu_NUM) ).

thf(13695,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1733]) ).

thf(435,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_LID) ).

thf(5884,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
      = ( s @ ( cart @ real @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).

thf(2203,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ ( fun @ B @ ( fun @ C @ A ) ) ) @ curry ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ C @ F ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ D ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ E ) ) ) @ ( s @ C @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCURRYu_DEF) ).

thf(16494,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ B @ C ) @ A ) @ ( fun @ B @ ( fun @ C @ A ) ) ) @ curry ) @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ C @ F ) ) )
      = ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ B @ C ) @ A ) @ D ) @ ( s @ ( prod @ B @ C ) @ ( i @ ( s @ ( fun @ C @ ( prod @ B @ C ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( prod @ B @ C ) ) ) @ c_ ) @ ( s @ B @ E ) ) ) @ ( s @ C @ F ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2203]) ).

thf(372,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
          <=> ( ( s @ num @ C )
              = ( s @ num @ B ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBASISu_COMPONENT) ).

thf(5477,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ? [D: $i] :
          ( ( ( p @ ( s @ bool @ D ) )
           => ( ( s @ num @ C )
              = ( s @ num @ B ) ) )
          & ( ( ( s @ num @ C )
              = ( s @ num @ B ) )
           => ( p @ ( s @ bool @ D ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).

thf(1800,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_RMUL) ).

thf(14208,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1800]) ).

thf(381,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ B @ b0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ B @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( ( s @ ( fun @ A @ bool ) @ s0 )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
        & ( ( s @ B @ a0 )
          = ( s @ B @ b0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINRECu_conjunct0) ).

thf(5527,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ B @ b0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ B @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( ( s @ ( fun @ A @ bool ) @ s0 )
            = ( s @ ( fun @ A @ bool ) @ empty ) )
          & ( ( s @ B @ a0 )
            = ( s @ B @ b0 ) ) ) )
      & ( ( ( ( s @ ( fun @ A @ bool ) @ s0 )
            = ( s @ ( fun @ A @ bool ) @ empty ) )
          & ( ( s @ B @ a0 )
            = ( s @ B @ b0 ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ ( fun @ num @ bool ) ) ) ) ) @ finrec ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ f ) ) ) @ ( s @ B @ b0 ) ) ) @ ( s @ ( fun @ A @ bool ) @ s0 ) ) ) @ ( s @ B @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).

thf(998,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUB) ).

thf(9051,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[998]) ).

thf(1908,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aABSORPTION) ).

thf(14814,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ C ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ ( fun @ A @ bool ) @ C ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1908]) ).

thf(1378,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_HALFu_conjunct2) ).

thf(11457,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1378]) ).

thf(1754,axiom,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
        = ( s @ A @ B ) )
      & ! [E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ E ) ) ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ A ) ) @ C ) @ ( s @ num @ E ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_RECURSIONu_STD) ).

thf(13862,plain,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
        = ( s @ A @ B ) )
      & ! [E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ E ) ) ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ A @ A ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ A ) ) @ C ) @ ( s @ num @ E ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1754]) ).

thf(279,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EXPu_conjunct7) ).

thf(4885,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(304,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ A ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_FINITEu_IMAGE) ).

thf(5014,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ A ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ univ ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(2406,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_ADDR) ).

thf(17561,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum0 @ A @ B ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2406]) ).

thf(506,axiom,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) )
      | ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',adestu_intu_rep) ).

thf(6226,plain,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) )
      | ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).

thf(1177,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ C ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_ADDu_RDISTRIB) ).

thf(10054,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ C ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1177]) ).

thf(894,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_LT) ).

thf(8489,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[894]) ).

thf(1105,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) )
         => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_OFFSETu_0) ).

thf(9596,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) )
         => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) )
            = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( fun @ num @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ B ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1105]) ).

thf(1133,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',amatrixu_cmul) ).

thf(9844,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ real @ D ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) )
              = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) @ ( s @ num @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) )
              = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ lambda ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ num @ ( cart @ real @ A ) ) ) ) @ D ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1133]) ).

thf(1264,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_INVu_LTu_1) ).

thf(10590,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1264]) ).

thf(2176,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_TRIANGLEu_SUB) ).

thf(16341,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2176]) ).

thf(3034,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
    <=> ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_REDUNDANTu_EQ) ).

thf(20952,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      & ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3034]) ).

thf(1660,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_CROSS) ).

thf(13294,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ cross0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1660]) ).

thf(533,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_NEGR) ).

thf(6366,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).

thf(1735,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_NUMSEGu_LE) ).

thf(13701,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1735]) ).

thf(3149,axiom,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_ANTISYM) ).

thf(21594,plain,
    ! [A: $i,B: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3149]) ).

thf(802,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        <=> ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIFFu_UNIONS) ).

thf(7955,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) ) )
          & ( ? [F: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ B ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[802]) ).

thf(1208,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) )
          = ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anaddu_add) ).

thf(10250,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) @ ( s @ num @ D ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) )
          = ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ ( fun @ num @ num ) ) ) @ A ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1208]) ).

thf(570,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_maxu_th) ).

thf(6600,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_max ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_max ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).

thf(841,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
            <=> ? [H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ I ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ H ) ) ) @ ( s @ A @ I ) ) )
                  = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ H ) @ ( s @ A @ I ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                 => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ H ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ I ) ) ) ) )
                    = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ H ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ I ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_CASES) ).

thf(8207,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) ) )
          & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) )
               => ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) ) )
              & ( ? [H: $i,I: $i] :
                    ( ( ( p @ ( s @ bool @ I ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ H ) ) ) ) )
                     => ( p @ ( s @ bool @ I ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ A @ H ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ H ) ) ) @ ( s @ A @ I ) ) )
                  = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ F ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ G ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ H ) @ ( s @ A @ I ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                 => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( fun @ A @ ( cart @ real @ B ) ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ H ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ I ) ) ) ) )
                    = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ H ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ I ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[841]) ).

thf(195,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_EQu_0) ).

thf(4281,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(3049,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_CMULu_ADDu_LDISTRIB) ).

thf(21080,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ E ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3049]) ).

thf(3096,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
    ? [F: $i] :
      ( ( ( p @ ( s @ bool @ F ) )
      <=> ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ G ) ) ) ) )
      & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ u_seqpattern ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ B @ E ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',au_SEQPATTERN) ).

thf(21344,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
    ? [F: $i] :
      ( ( ( p @ ( s @ bool @ F ) )
       => ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ G ) ) ) ) )
      & ( ? [G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ G ) ) ) )
       => ( p @ ( s @ bool @ F ) ) )
      & ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ u_seqpattern ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ B @ E ) ) )
        = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ cond ) @ ( s @ bool @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3096]) ).

thf(2628,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_HULL) ).

thf(18776,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2628]) ).

thf(2470,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_RADD) ).

thf(17913,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2470]) ).

thf(389,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) )
          = ( s @ bool @ t ) )
     => ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot )
        = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ h_ ) @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZBOT) ).

thf(5559,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) )
          = ( s @ bool @ t ) )
     => ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ zbot )
        = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ injn ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ h_ ) @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).

thf(2547,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_SUB2) ).

thf(18342,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2547]) ).

thf(2049,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_TRANS) ).

thf(15647,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2049]) ).

thf(1235,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ union ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_COMBINEu_L) ).

thf(10408,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ union ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1235]) ).

thf(3012,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ E ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ E ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_EQu_NUMSEG) ).

thf(20838,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ D ) ) ) ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ E ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ E ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3012]) ).

thf(697,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_ADDu_LDISTRIB) ).

thf(7323,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ C ) ) ) ) )
      = ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_add ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) ) ) @ ( s @ hreal @ ( i @ ( s @ ( fun @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ hreal ) ) @ hrealu_mul ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[697]) ).

thf(904,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_sgn) ).

thf(8542,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[904]) ).

thf(1451,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ num @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVISIONu_SIMPu_conjunct0) ).

thf(11943,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ num @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1451]) ).

thf(717,axiom,
    ! [A: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SGNu_EQu_conjunct1) ).

thf(7433,plain,
    ! [A: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_gt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[717]) ).

thf(1813,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ B )
        = ( s @ ( fun @ A @ bool ) @ C ) )
    <=> ! [D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXTENSION) ).

thf(14266,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ C ) )
       => ! [D: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
      & ( ! [D: $i] :
            ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
            = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1813]) ).

thf(3036,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ surj ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
    <=> ( ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) )
        & ! [F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
           => ? [G: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) )
                  = ( s @ B @ F ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJ) ).

thf(20962,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ surj ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
       => ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) )
          & ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
             => ? [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ F ) ) ) ) ) )
      & ( ( ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) )
          & ! [F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
             => ? [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ F ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ surj ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3036]) ).

thf(2134,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
        ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) )
    <=> ? [D: $i] :
        ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSKOLEMu_THM) ).

thf(16113,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
          ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) )
       => ? [D: $i] :
          ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) ) ) )
      & ( ? [D: $i] :
          ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) @ ( s @ B @ E ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) ) ) ) )
       => ! [D: $i] :
          ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ C ) @ ( s @ B @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2134]) ).

thf(2456,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_SYM) ).

thf(17839,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2456]) ).

thf(2601,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_TRIANGLEu_LT) ).

thf(18638,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2601]) ).

thf(951,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ G ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
             => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) @ ( s @ num @ F ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLINEARu_COMPONENTWISE) ).

thf(8833,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ num @ G ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ E ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ G ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ bool ) @ linear ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) )
             => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) ) ) @ ( s @ num @ F ) ) )
                = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ ( fun @ num @ ( fun @ num @ real ) ) ) ) @ C ) @ ( s @ ( cart @ real @ A ) @ E ) ) ) @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ B ) ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[951]) ).

thf(2476,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_ADDu_RCANCEL) ).

thf(17937,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2476]) ).

thf(2911,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_INCLu_EXCL) ).

thf(20275,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2911]) ).

thf(870,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EXPu_conjunct9) ).

thf(8400,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[870]) ).

thf(1305,axiom,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_1) ).

thf(10809,plain,
    ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
    = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1305]) ).

thf(1817,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        <=> ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCANTORu_THM) ).

thf(14288,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
          & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
     => ! [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1817]) ).

thf(1634,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADDu_SUB2) ).

thf(13101,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1634]) ).

thf(3162,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ C ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_ADDNEG) ).

thf(21653,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ C ) ) ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3162]) ).

thf(491,axiom,
    ! [A: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_ZERO) ).

thf(6169,plain,
    ! [A: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).

thf(735,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_SUBSET) ).

thf(7557,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ D ) @ ( s @ A @ E ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[735]) ).

thf(557,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_min) ).

thf(6520,plain,
    ! [A: $i,B: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).

thf(1515,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_UNIONu_RZERO) ).

thf(12307,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1515]) ).

thf(164,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_GT0) ).

thf(4007,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(3067,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ B ) ) ) @ ( s @ A @ D ) ) ) )
    <=> ( ( p @ ( s @ bool @ B ) )
        & ( ( s @ A @ C )
          = ( s @ A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSETSPEC) ).

thf(21177,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ B ) ) ) @ ( s @ A @ D ) ) ) )
       => ( ( p @ ( s @ bool @ B ) )
          & ( ( s @ A @ C )
            = ( s @ A @ D ) ) ) )
      & ( ( ( p @ ( s @ bool @ B ) )
          & ( ( s @ A @ C )
            = ( s @ A @ D ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ C ) ) ) @ ( s @ bool @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3067]) ).

thf(3168,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHULLu_UNIONu_RIGHT) ).

thf(21690,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ hull ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3168]) ).

thf(1802,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ u_unguardedu_pattern ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) ) )
    <=> ( ( p @ ( s @ bool @ A ) )
        & ( p @ ( s @ bool @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',au_UNGUARDEDu_PATTERN) ).

thf(14214,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ u_unguardedu_pattern ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) ) )
       => ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ B ) ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          & ( p @ ( s @ bool @ B ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ u_unguardedu_pattern ) @ ( s @ bool @ A ) ) ) @ ( s @ bool @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1802]) ).

thf(648,axiom,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ bool @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALLu_conjunct0) ).

thf(7050,plain,
    ! [A: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ bool @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[648]) ).

thf(3033,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ sing ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
    <=> ? [C: $i] :
          ( ( s @ ( fun @ A @ bool ) @ B )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSING) ).

thf(20945,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ sing ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
       => ? [C: $i] :
            ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) ) )
      & ( ? [C: $i] :
            ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ sing ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3033]) ).

thf(2187,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_INFINITE) ).

thf(16407,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ real @ bool ) @ univ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2187]) ).

thf(169,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) )
       != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBASISu_NONZERO) ).

thf(4024,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) )
       != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(2455,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GEu_conjunct6) ).

thf(17836,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2455]) ).

thf(2902,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ indu_0 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMu_REPu_RULESu_conjunct0) ).

thf(20243,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ind0 @ bool ) @ numu_rep ) @ ( s @ ind0 @ indu_0 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2902]) ).

thf(1385,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ C @ H ) ) ) )
        <=> ? [I: $i,J: $i,K: $i] :
              ( ( ( p @ ( s @ bool @ K ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) @ gspec ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_PRODUCTu_DEPENDENT) ).

thf(11522,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ C @ H ) ) ) )
           => ? [I: $i,J: $i,K: $i] :
                ( ( ( p @ ( s @ bool @ K ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ K ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) ) )
          & ( ? [I: $i,J: $i,K: $i] :
                ( ( ( p @ ( s @ bool @ K ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ K ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ C @ ( fun @ bool @ ( fun @ C @ bool ) ) ) @ setspec ) @ ( s @ C @ H ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) @ ( s @ A @ I ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) )
     => ! [E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            & ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ ( fun @ C @ bool ) ) @ gspec ) @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ C @ bool ) ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ E ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1385]) ).

thf(2469,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_ge ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_ge) ).

thf(17910,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_ge ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_ge ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ int @ real ) @ realu_ofu_int ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2469]) ).

thf(1139,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
     => ? [C: $i] :
          ( ( ( p @ ( s @ bool @ C ) )
          <=> ( ( s @ ( fun @ real @ bool ) @ B )
              = ( s @ ( fun @ real @ bool ) @ empty ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFu_INSERTu_FINITE) ).

thf(9874,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
     => ? [C: $i] :
          ( ( ( p @ ( s @ bool @ C ) )
           => ( ( s @ ( fun @ real @ bool ) @ B )
              = ( s @ ( fun @ real @ bool ) @ empty ) ) )
          & ( ( ( s @ ( fun @ real @ bool ) @ B )
              = ( s @ ( fun @ real @ bool ) @ empty ) )
           => ( p @ ( s @ bool @ C ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_min ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1139]) ).

thf(2241,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_ASSOC) ).

thf(16671,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2241]) ).

thf(2288,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) )
    <=> ( ( s @ num @ A )
        = ( s @ num @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_EQu_conjunct9) ).

thf(16901,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) )
       => ( ( s @ num @ A )
          = ( s @ num @ B ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ B ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ A ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2288]) ).

thf(1725,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXPu_ADD) ).

thf(13657,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1725]) ).

thf(1727,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_2) ).

thf(13663,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ A ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1727]) ).

thf(592,axiom,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_GTu_conjunct2) ).

thf(6765,plain,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ g_ ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).

thf(394,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) )
     => ( ( s @ num @ B )
        = ( s @ num @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBASISu_INJ) ).

thf(5586,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) )
        & ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) )
     => ( ( s @ num @ B )
        = ( s @ num @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).

thf(3005,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ delete ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ B @ D ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ delete ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPPORTu_CLAUSESu_conjunct2) ).

thf(20799,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ delete ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ B @ D ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ delete ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) @ ( s @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3005]) ).

thf(432,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_UNIONu_NONZERO) ).

thf(5867,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
           => ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).

thf(2516,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_NOTu_LE) ).

thf(18142,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_u_c ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) )
       => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2516]) ).

thf(1535,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_NEGu_MINUS1) ).

thf(12476,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1535]) ).

thf(1611,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ num @ A ) @ B )
      = ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) ) @ fcons ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ A ) ) ) @ o ) @ ( s @ ( fun @ num @ A ) @ B ) ) ) @ ( s @ ( fun @ num @ num ) @ suc ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFCONSu_UNDO) ).

thf(12919,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ num @ A ) @ B )
      = ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ A ) @ ( fun @ num @ A ) ) ) @ fcons ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ num ) @ ( fun @ num @ A ) ) ) @ o ) @ ( s @ ( fun @ num @ A ) @ B ) ) ) @ ( s @ ( fun @ num @ num ) @ suc ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1611]) ).

thf(361,axiom,
    ( ( ( p @ ( s @ bool @ p1 ) )
      & ( p @ ( s @ bool @ q ) )
      & ( p @ ( s @ bool @ r0 ) ) )
  <=> ( ( p @ ( s @ bool @ q ) )
      & ( p @ ( s @ bool @ p1 ) )
      & ( p @ ( s @ bool @ r0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONJu_ACIu_conjunct2) ).

thf(5378,plain,
    ( ( ( ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ q ) )
        & ( p @ ( s @ bool @ r0 ) ) )
     => ( ( p @ ( s @ bool @ q ) )
        & ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ r0 ) ) ) )
    & ( ( ( p @ ( s @ bool @ q ) )
        & ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ r0 ) ) )
     => ( ( p @ ( s @ bool @ p1 ) )
        & ( p @ ( s @ bool @ q ) )
        & ( p @ ( s @ bool @ r0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).

thf(1402,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ card ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_NUMSEGu_LE) ).

thf(11668,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ num ) @ card ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1402]) ).

thf(1917,axiom,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
           => ( ? [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                  & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) )
                    = ( s @ A @ C ) ) )
              & ! [D: $i,E: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) )
                      = ( s @ A @ C ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ E ) ) )
                      = ( s @ A @ C ) ) )
                 => ( ( s @ B @ D )
                    = ( s @ B @ E ) ) ) ) ) )
    <=> ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) )
        & ? [C: $i] :
            ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) )
            & ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ D ) ) ) ) )
                  = ( s @ A @ D ) ) )
            & ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
               => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
                  = ( s @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBIJECTIVEu_INVERSES) ).

thf(14888,plain,
    ! [A: $i,B: $i] :
      ( ( ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
             => ( ? [D: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) )
                      = ( s @ A @ C ) ) )
                & ! [D: $i,E: $i] :
                    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) )
                        = ( s @ A @ C ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ E ) ) )
                        = ( s @ A @ C ) ) )
                   => ( ( s @ B @ D )
                      = ( s @ B @ E ) ) ) ) ) )
       => ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) )
          & ? [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                 => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ D ) ) ) ) )
                    = ( s @ A @ D ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                 => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
                    = ( s @ B @ D ) ) ) ) ) )
      & ( ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) )
          & ? [C: $i] :
              ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
                 => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ D ) ) ) ) )
                    = ( s @ A @ D ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                 => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) ) ) )
                    = ( s @ B @ D ) ) ) ) )
       => ( ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) ) )
          & ! [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ t0 ) ) ) )
             => ( ? [D: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                    & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) )
                      = ( s @ A @ C ) ) )
                & ! [D: $i,E: $i] :
                    ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ D ) ) )
                        = ( s @ A @ C ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ f ) @ ( s @ B @ E ) ) )
                        = ( s @ A @ C ) ) )
                   => ( ( s @ B @ D )
                      = ( s @ B @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1917]) ).

thf(2737,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ univ ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_FUNSPACEu_CURRY) ).

thf(19313,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ ( fun @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ ( fun @ ( prod @ A @ B ) @ C ) @ bool ) @ univ ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2737]) ).

thf(2231,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ C @ bool ) @ univ ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) @ ( s @ ( fun @ D @ bool ) @ univ ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ bool ) @ univ ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_FUNSPACEu_LE) ).

thf(16642,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ C @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ C @ bool ) @ univ ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) @ ( s @ ( fun @ D @ bool ) @ univ ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( fun @ A @ B ) @ bool ) @ univ ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ bool ) @ univ ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2231]) ).

thf(1575,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_SUBu_conjunct6) ).

thf(12707,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1575]) ).

thf(2774,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ D @ H ) ) ) )
        <=> ? [I: $i,J: $i] :
              ( ( ( p @ ( s @ bool @ J ) )
              <=> ! [K: $i,L: $i,M: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ in ) @ ( s @ D @ I ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ bool @ ( fun @ D @ bool ) ) ) @ setspec ) @ ( s @ D @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ D @ I ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) )
            <=> ? [J: $i,K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) )
         => ! [G: $i,H: $i] :
              ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( fun @ D @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) ) ) ) )
              = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ D @ bool ) ) @ gspec ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERSu_GSPECu_conjunct2) ).

thf(19541,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ D @ H ) ) ) )
           => ? [I: $i,J: $i] :
                ( ( ( p @ ( s @ bool @ J ) )
                 => ! [K: $i,L: $i,M: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ in ) @ ( s @ D @ I ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) ) ) )
                & ( ! [K: $i,L: $i,M: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ in ) @ ( s @ D @ I ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ J ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ bool @ ( fun @ D @ bool ) ) ) @ setspec ) @ ( s @ D @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ D @ I ) ) ) ) ) )
          & ( ? [I: $i,J: $i] :
                ( ( ( p @ ( s @ bool @ J ) )
                 => ! [K: $i,L: $i,M: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ in ) @ ( s @ D @ I ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) ) ) )
                & ( ! [K: $i,L: $i,M: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ in ) @ ( s @ D @ I ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ A @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ M ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ J ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ bool @ ( fun @ D @ bool ) ) ) @ setspec ) @ ( s @ D @ H ) ) ) @ ( s @ bool @ J ) ) ) @ ( s @ D @ I ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ D @ H ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) )
               => ? [J: $i,K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) ) )
              & ( ? [J: $i,K: $i,L: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) @ ( s @ A @ J ) ) ) @ ( s @ B @ K ) ) ) @ ( s @ C @ L ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ D @ bool ) @ I ) ) ) ) ) )
         => ! [G: $i,H: $i] :
              ( ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( fun @ D @ bool ) ) @ inters ) @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ D @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ D @ bool ) @ bool ) ) ) @ F ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) ) ) ) )
              = ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ bool ) @ ( fun @ D @ bool ) ) @ gspec ) @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ D @ bool ) ) ) @ E ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ bool ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2774]) ).

thf(2810,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
        <=> ( ( s @ A @ D )
            = ( s @ A @ C ) ) )
     => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
        = ( s @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSELECTu_UNIQUE) ).

thf(19720,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
           => ( ( s @ A @ D )
              = ( s @ A @ C ) ) )
          & ( ( ( s @ A @ D )
              = ( s @ A @ C ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) )
     => ( ( s @ A @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ A ) @ h_ ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
        = ( s @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2810]) ).

thf(82,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_DIFF) ).

thf(3633,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(1470,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
     => ? [C: $i] :
          ( ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) )
          & ! [D: $i,E: $i] :
              ( ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ D ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ E ) ) ) )
             => ( ( s @ num @ D )
                = ( s @ num @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_ASu_INJECTIVEu_IMAGE) ).

thf(12024,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ) )
     => ? [C: $i] :
          ( ( ( s @ ( fun @ A @ bool ) @ B )
            = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ A ) @ ( fun @ ( fun @ num @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ num @ A ) @ C ) ) ) @ ( s @ ( fun @ num @ bool ) @ univ ) ) ) )
          & ! [D: $i,E: $i] :
              ( ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ D ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ C ) @ ( s @ num @ E ) ) ) )
             => ( ( s @ num @ D )
                = ( s @ num @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1470]) ).

thf(1488,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_MULTu_RMOD) ).

thf(12121,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1488]) ).

thf(1328,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_2) ).

thf(11028,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1328]) ).

thf(2909,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ A ) @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIMINDEXu_FINITEu_IMAGE) ).

thf(20265,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ A ) @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2909]) ).

thf(1867,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SGNu_NEG) ).

thf(14563,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1867]) ).

thf(1141,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_0) ).

thf(9883,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1141]) ).

thf(2694,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
      = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_SUPPORT) ).

thf(19096,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) )
      = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2694]) ).

thf(2673,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFINITEu_DIFFu_FINITE) ).

thf(18980,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2673]) ).

thf(1043,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_SUB) ).

thf(9293,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ y ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ y ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1043]) ).

thf(2785,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ F ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ E ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i] :
              ( ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ G ) ) )
              = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ G ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) ) ) )
                & ! [J: $i,K: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ B @ K ) ) ) )
                   => ! [L: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) @ ( s @ C @ L ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) @ ( s @ C @ L ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ H ) @ ( s @ B @ K ) ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPERADMISSIBLEu_TAIL) ).

thf(19601,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ E ) ) ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ F ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ E ) @ ( s @ ( fun @ C @ A ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i] :
              ( ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ G ) ) )
              = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ G ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) ) ) )
                & ! [J: $i,K: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ B @ K ) ) ) )
                   => ! [L: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) @ ( s @ C @ L ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ B @ K ) ) ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) @ ( s @ C @ L ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ H ) @ ( s @ B @ K ) ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2785]) ).

thf(1034,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ empty ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_DISJOINTu_UNION) ).

thf(9250,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
        = ( s @ ( fun @ A @ bool ) @ empty ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ ( sum0 @ A @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum0 @ A @ A ) @ bool ) ) ) @ p_u_c ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1034]) ).

thf(624,axiom,
    ! [A: $i] :
      ( ? [B: $i] :
          ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) )
      | ? [B: $i] :
          ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_IMAGE) ).

thf(6952,plain,
    ! [A: $i] :
      ( ? [B: $i] :
          ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) )
      | ? [B: $i] :
          ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).

thf(2706,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( s @ A @ D )
            = ( s @ A @ C ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_UNWINDu_THM2) ).

thf(19161,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( ( s @ A @ D )
              = ( s @ A @ C ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
       => ! [D: $i] :
            ( ( ( s @ A @ D )
              = ( s @ A @ C ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2706]) ).

thf(1896,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTERu_EMPTYu_conjunct1) ).

thf(14717,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) )
      = ( s @ ( fun @ A @ bool ) @ empty ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1896]) ).

thf(327,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i] :
        ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LINVu_LEMMA3) ).

thf(5200,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i] :
        ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).

thf(1097,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADD1) ).

thf(9564,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1097]) ).

thf(2484,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
          | ( p @ ( s @ bool @ C ) ) )
    <=> ( ! [D: $i] : ( p @ ( s @ bool @ B ) )
        | ! [D: $i] : ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRIVu_FORALLu_ORu_THM) ).

thf(17978,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2484]) ).

thf(1164,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ? [D: $i] :
        ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUPPERu_BOUNDu_FINITEu_SET) ).

thf(9987,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ? [D: $i] :
        ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1164]) ).

thf(2812,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( ( s @ A @ C )
         != ( s @ A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINu_DELETE) ).

thf(19733,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ( ( s @ A @ C )
           != ( s @ A @ D ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
          & ( ( s @ A @ C )
           != ( s @ A @ D ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ A @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2812]) ).

thf(2976,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) )
      = ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ univ ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULu_Cu_UNIV) ).

thf(20616,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( prod @ A @ B ) @ bool ) ) ) @ t_u_c ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) )
      = ( s @ ( fun @ ( prod @ A @ B ) @ bool ) @ univ ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2976]) ).

thf(1317,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBu_ADDu_LCANCEL) ).

thf(10981,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1317]) ).

thf(774,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) )
    <=> ( ( s @ ( cart @ real @ A ) @ B )
       != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_POSu_LT) ).

thf(7819,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
         != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ ( cart @ real @ A ) @ B )
         != ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[774]) ).

thf(3065,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ A )
        = ( s @ int @ B ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_TOTAL) ).

thf(21172,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ A )
        = ( s @ int @ B ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3065]) ).

thf(1481,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ univ ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIMINDEXu_FINITEu_SUM) ).

thf(12102,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ ( finite_sum @ A @ B ) @ bool ) @ univ ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1481]) ).

thf(2676,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ A @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDELETEu_COMM) ).

thf(18987,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ A @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ C ) ) ) ) ) @ ( s @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2676]) ).

thf(2930,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_LMUL) ).

thf(20397,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2930]) ).

thf(2121,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_IDEMPOT) ).

thf(16021,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2121]) ).

thf(845,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        | ( ( s @ num @ B )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_EQu_0) ).

thf(8236,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          | ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          | ( ( s @ num @ B )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[845]) ).

thf(1187,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_DIV) ).

thf(10092,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1187]) ).

thf(928,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( ( s @ num @ C )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLTu_MULTu_RCANCEL) ).

thf(8692,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          & ( ( s @ num @ C )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
          & ( ( s @ num @ C )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[928]) ).

thf(2289,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        | ( ( s @ real @ A )
          = ( s @ real @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_LT) ).

thf(16908,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
          | ( ( s @ real @ A )
            = ( s @ real @ B ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
          | ( ( s @ real @ A )
            = ( s @ real @ B ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2289]) ).

thf(880,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ num @ B ) ) @ C ) @ ( s @ ( list @ B ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ ( list @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ B ) @ B ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( list @ B ) @ D ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( cart @ B @ A ) ) @ vector ) @ ( s @ ( list @ B ) @ D ) ) )
          = ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( cart @ B @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ num @ B ) ) @ C ) @ ( s @ ( list @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',avector) ).

thf(8437,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ num @ B ) ) @ C ) @ ( s @ ( list @ B ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ B @ ( i @ ( s @ ( fun @ ( list @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( list @ B ) @ B ) ) @ el ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( list @ B ) @ D ) ) ) )
     => ! [D: $i] :
          ( ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( cart @ B @ A ) ) @ vector ) @ ( s @ ( list @ B ) @ D ) ) )
          = ( s @ ( cart @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ B ) @ ( cart @ B @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ B ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( fun @ num @ B ) ) @ C ) @ ( s @ ( list @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[880]) ).

thf(1939,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ bool ) @ isu_nadd ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aisu_naddu_0) ).

thf(15059,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ A ) @ ( s @ num @ B ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ bool ) @ isu_nadd ) @ ( s @ ( fun @ num @ num ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1939]) ).

thf(960,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_CONVEXu_BOUNDu_LT) ).

thf(8857,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
        & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ B ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[960]) ).

thf(2501,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ decimal ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDECIMAL) ).

thf(18042,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ decimal ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_div ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2501]) ).

thf(3122,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISJOINTu_INSERT) ).

thf(21475,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
          & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3122]) ).

thf(528,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFACTu_conjunct1) ).

thf(6351,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ fact ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).

thf(2025,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_MULu_WELLDEFR) ).

thf(15553,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2025]) ).

thf(594,axiom,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
      <=> ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ A ) @ fst ) @ ( s @ ( prod @ A @ B ) @ h0 ) ) )
          = ( s @ A @ a0 ) ) )
      & ( ( s @ B @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) ) @ assoc ) @ ( s @ A @ a0 ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) ) @ cons ) @ ( s @ ( prod @ A @ B ) @ h0 ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ t0 ) ) ) ) )
        = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ B ) @ snd ) @ ( s @ ( prod @ A @ B ) @ h0 ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) ) @ assoc ) @ ( s @ A @ a0 ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ t0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aASSOC) ).

thf(6771,plain,
    ! [A: $i,B: $i] :
    ? [C: $i] :
      ( ( ( p @ ( s @ bool @ C ) )
       => ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ A ) @ fst ) @ ( s @ ( prod @ A @ B ) @ h0 ) ) )
          = ( s @ A @ a0 ) ) )
      & ( ( ( s @ A @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ A ) @ fst ) @ ( s @ ( prod @ A @ B ) @ h0 ) ) )
          = ( s @ A @ a0 ) )
       => ( p @ ( s @ bool @ C ) ) )
      & ( ( s @ B @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) ) @ assoc ) @ ( s @ A @ a0 ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ ( fun @ ( list @ ( prod @ A @ B ) ) @ ( list @ ( prod @ A @ B ) ) ) ) @ cons ) @ ( s @ ( prod @ A @ B ) @ h0 ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ t0 ) ) ) ) )
        = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ C ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( prod @ A @ B ) @ B ) @ snd ) @ ( s @ ( prod @ A @ B ) @ h0 ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ ( prod @ A @ B ) ) @ B ) ) @ assoc ) @ ( s @ A @ a0 ) ) ) @ ( s @ ( list @ ( prod @ A @ B ) ) @ t0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).

thf(676,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ h0 ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEXu_conjunct1) ).

thf(7214,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ h0 ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ p0 ) @ ( s @ A @ h0 ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ p0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[676]) ).

thf(1466,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
            = ( s @ A @ B ) )
          & ! [E: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ E ) ) ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ A ) ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ E ) ) ) ) )
      & ! [D: $i,E: $i] :
          ( ( ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
              = ( s @ A @ B ) )
            & ! [F: $i] :
                ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ F ) ) ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ A ) ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ F ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
              = ( s @ A @ B ) )
            & ! [F: $i] :
                ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ F ) ) ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ A ) ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ E ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ F ) ) ) ) )
         => ( ( s @ ( fun @ num @ A ) @ D )
            = ( s @ ( fun @ num @ A ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_Axiom) ).

thf(12002,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
            = ( s @ A @ B ) )
          & ! [E: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ E ) ) ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ A ) ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ E ) ) ) ) )
      & ! [D: $i,E: $i] :
          ( ( ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
              = ( s @ A @ B ) )
            & ! [F: $i] :
                ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ F ) ) ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ A ) ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ F ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
              = ( s @ A @ B ) )
            & ! [F: $i] :
                ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ E ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ F ) ) ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ A @ ( fun @ num @ A ) ) @ C ) @ ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ E ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ F ) ) ) ) )
         => ( ( s @ ( fun @ num @ A ) @ D )
            = ( s @ ( fun @ num @ A ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1466]) ).

thf(450,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( finite_sum @ A @ B ) @ ( i @ ( s @ ( fun @ num @ ( finite_sum @ A @ B ) ) @ mku_finiteu_sum ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( finite_sum @ A @ B ) @ num ) @ destu_finiteu_sum ) @ ( s @ ( finite_sum @ A @ B ) @ C ) ) ) ) )
      = ( s @ ( finite_sum @ A @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',afiniteu_sumu_tybiju_conjunct0) ).

thf(5945,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( finite_sum @ A @ B ) @ ( i @ ( s @ ( fun @ num @ ( finite_sum @ A @ B ) ) @ mku_finiteu_sum ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( finite_sum @ A @ B ) @ num ) @ destu_finiteu_sum ) @ ( s @ ( finite_sum @ A @ B ) @ C ) ) ) ) )
      = ( s @ ( finite_sum @ A @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(497,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNUMSEGu_REC) ).

thf(6190,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) )
     => ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) )
        = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) ) @ insert ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).

thf(378,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ B ) ) )
          = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',afrom) ).

thf(5500,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ B ) ) )
          = ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).

thf(1523,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_LADDu_0) ).

thf(12406,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ num @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1523]) ).

thf(2701,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
        | ( ( s @ int @ A )
          = ( s @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_LT) ).

thf(19129,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
          | ( ( s @ int @ A )
            = ( s @ int @ B ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) )
          | ( ( s @ int @ A )
            = ( s @ int @ B ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2701]) ).

thf(2683,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_UNION) ).

thf(19036,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ disjoint ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
     => ( ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) )
        = ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ B ) ) @ ( cart @ real @ B ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ B ) ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2683]) ).

thf(2326,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( ? [J: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) )
                    = ( s @ B @ I ) ) )
              & ! [J: $i,K: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) )
                      = ( s @ B @ I ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ K ) ) )
                      = ( s @ B @ I ) ) )
                 => ( ( s @ A @ J )
                    = ( s @ A @ K ) ) ) ) )
        & ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
              & ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ I ) ) ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ F ) ) )
        = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_EQu_GENERAL) ).

thf(17056,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ I ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ( ? [J: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) )
                    = ( s @ B @ I ) ) )
              & ! [J: $i,K: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ J ) ) )
                      = ( s @ B @ I ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ K ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ K ) ) )
                      = ( s @ B @ I ) ) )
                 => ( ( s @ A @ J )
                    = ( s @ A @ K ) ) ) ) )
        & ! [I: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ I ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
              & ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ H ) @ ( s @ A @ I ) ) ) ) )
                = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ F ) @ ( s @ A @ I ) ) ) ) ) ) )
     => ( ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( cart @ real @ C ) ) @ F ) ) )
        = ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ C ) ) @ ( cart @ real @ C ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ C ) ) @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2326]) ).

thf(2349,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SGNu_ABS) ).

thf(17172,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2349]) ).

thf(3017,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ C @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ F ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i] :
              ( ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ G ) ) )
              = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ C @ A ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ! [J: $i,K: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ B @ K ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ I ) @ ( s @ B @ K ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ H ) @ ( s @ B @ K ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_BASE) ).

thf(20859,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ C @ A ) @ E ) ) ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) @ ( s @ B @ G ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ C @ A ) @ E ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ F ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [E: $i] :
          ( ! [F: $i,G: $i] :
              ( ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) @ ( s @ ( fun @ C @ A ) @ G ) ) )
              = ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ B @ A ) ) ) @ D ) @ ( s @ ( fun @ C @ A ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ F ) ) ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ! [J: $i,K: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) @ ( s @ ( fun @ C @ A ) @ J ) ) ) @ ( s @ B @ K ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ I ) @ ( s @ B @ K ) ) ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ H ) @ ( s @ B @ K ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ C ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ B @ A ) ) ) @ E ) @ ( s @ ( fun @ B @ C ) @ I ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3017]) ).

thf(1824,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ? [D: $i] :
        ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUPPERu_BOUNDu_FINITEu_SETu_REAL) ).

thf(14325,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ? [D: $i] :
        ! [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ B ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ real @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1824]) ).

thf(2074,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aZRECSPACEu_RULESu_conjunct1) ).

thf(15737,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ D ) @ ( s @ num @ E ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ bool ) @ zrecspace ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) ) @ zconstr ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2074]) ).

thf(784,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_CLAUSESu_conjunct1) ).

thf(7880,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).

thf(1528,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_RID) ).

thf(12435,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ vec ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1528]) ).

thf(589,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_EQu_ABS) ).

thf(6750,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ num @ A )
         != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        & ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ C ) ) ) @ ( s @ num @ A ) ) ) ) )
     => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).

thf(1787,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_REFL) ).

thf(14127,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1787]) ).

thf(2568,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
    <=> ( ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
              & ( ( s @ A @ E )
               != ( s @ A @ C ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ E ) ) ) @ ( s @ A @ C ) ) ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIRWISEu_INSERT) ).

thf(18423,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
       => ( ! [E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( ( s @ A @ E )
                 != ( s @ A @ C ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ E ) ) ) @ ( s @ A @ C ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ( ! [E: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                & ( ( s @ A @ E )
                 != ( s @ A @ C ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ C ) ) ) @ ( s @ A @ E ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) @ ( s @ A @ E ) ) ) @ ( s @ A @ C ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ pairwise ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2568]) ).

thf(1766,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
    <=> ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_MOD) ).

thf(13937,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
       => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
      & ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1766]) ).

thf(2053,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_COUNTABLE) ).

thf(15662,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ B @ bool ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2053]) ).

thf(1599,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ B ) ) ) )
        <=> ? [C: $i,D: $i] :
              ( ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) )
                = ( s @ bool @ B ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ y1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ bool ) @ bool ) @ h_ ) @ ( s @ ( fun @ bool @ bool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_le) ).

thf(12855,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ B ) ) ) )
           => ? [C: $i,D: $i] :
                ( ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) )
                  = ( s @ bool @ B ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ y1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) ) )
          & ( ? [C: $i,D: $i] :
                ( ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_le ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) )
                  = ( s @ bool @ B ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ x1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ C ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ destu_real ) @ ( s @ real @ y1 ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ D ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ B ) ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ x1 ) ) ) @ ( s @ real @ y1 ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ bool @ bool ) @ bool ) @ h_ ) @ ( s @ ( fun @ bool @ bool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1599]) ).

thf(599,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_POSu_LE) ).

thf(6807,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).

thf(1884,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) )
        <=> ? [I: $i,J: $i,K: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) ) )
     => ! [F: $i,G: $i] :
          ( ! [H: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ H ) ) ) ) )
        <=> ! [H: $i,I: $i,J: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_INu_GSPECu_conjunct2) ).

thf(14651,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) )
           => ? [I: $i,J: $i,K: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) ) )
          & ( ? [I: $i,J: $i,K: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ H ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ I ) ) ) @ ( s @ C @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) @ ( s @ A @ H ) ) ) ) ) )
     => ! [F: $i,G: $i] :
          ( ( ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ H ) ) ) ) )
           => ! [H: $i,I: $i,J: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) ) ) )
          & ( ! [H: $i,I: $i,J: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ ( i @ ( s @ ( fun @ D @ A ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ A ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) @ ( s @ B @ H ) ) ) @ ( s @ C @ I ) ) ) @ ( s @ D @ J ) ) ) ) ) ) )
           => ! [H: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( fun @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ ( fun @ A @ bool ) ) ) @ E ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ F ) ) ) @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ A ) ) ) @ G ) ) ) ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ q0 ) @ ( s @ A @ H ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1884]) ).

thf(913,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_SQUAREu_ABS) ).

thf(8643,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[913]) ).

thf(400,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ real @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i,F: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ real @ E ) ) ) ) ) )
         => ! [C: $i] :
              ( ! [D: $i,E: $i,F: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
             => ! [D: $i,E: $i,F: $i,G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ G ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ E ) ) ) @ ( s @ real @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ B ) @ ( s @ num @ G ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_POLYFUNu_ALT) ).

thf(5637,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) ) @ ( s @ real @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [B: $i] :
          ( ! [C: $i,D: $i,E: $i,F: $i] :
              ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ num @ F ) ) ) @ ( s @ real @ E ) ) ) ) ) )
         => ! [C: $i] :
              ( ! [D: $i,E: $i,F: $i] :
                  ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) )
                  = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ D ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ E ) ) ) @ ( s @ num @ F ) ) ) ) ) )
             => ! [D: $i,E: $i,F: $i,G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ G ) ) ) )
                 => ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ E ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ G ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) ) ) )
                    = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ E ) ) ) @ ( s @ real @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ G ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ ( fun @ num @ real ) @ ( fun @ real @ ( fun @ num @ real ) ) ) ) @ B ) @ ( s @ num @ G ) ) ) @ ( s @ ( fun @ num @ real ) @ D ) ) ) @ ( s @ real @ F ) ) ) ) ) @ ( s @ real @ E ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).

thf(1805,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIFFu_DIFF) ).

thf(14225,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
      = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1805]) ).

thf(135,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',adot) ).

thf(3869,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(1320,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) )
        & ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( ( ( s @ num @ C )
          = ( s @ num @ E ) )
        & ( ( s @ num @ D )
          = ( s @ num @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVMODu_UNIQu_LEMMA) ).

thf(10988,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) )
        & ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ F ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ F ) ) ) @ ( s @ num @ B ) ) ) ) )
     => ( ( ( s @ num @ C )
          = ( s @ num @ E ) )
        & ( ( s @ num @ D )
          = ( s @ num @ F ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1320]) ).

thf(637,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
          & ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_BOUNDEDu_POS) ).

thf(6993,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ? [E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
          & ! [F: $i,G: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ E ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).

thf(179,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ A ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ univ ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_FUNSPACEu_UNIV) ).

thf(4118,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ A ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ univ ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ exp0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(2770,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_BETWEEN1) ).

thf(19528,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ B ) ) ) @ ( s @ real @ A ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2770]) ).

thf(813,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) )
    <=> ( ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_MULTu_LCANCEL) ).

thf(8034,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) )
       => ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
      & ( ( ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[813]) ).

thf(2072,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRANSPu_TRANSP) ).

thf(15732,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ B ) @ A ) ) @ transp ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2072]) ).

thf(579,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LNEGu_UNIQ) ).

thf(6697,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) ) )
      & ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_neg ) @ ( s @ int @ B ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).

thf(112,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_SUBSETu_IMAGE) ).

thf(3772,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) ) @ image ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(3088,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) )
      = ( s @ ( fun @ A @ bool ) @ univ ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINSERTu_UNIV) ).

thf(21277,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) )
      = ( s @ ( fun @ A @ bool ) @ univ ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3088]) ).

thf(3128,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
        ? [D: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
          = ( s @ B @ C ) )
    <=> ? [C: $i] :
        ! [D: $i] :
          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) ) ) )
          = ( s @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSURJECTIVEu_RIGHTu_INVERSE) ).

thf(21500,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
          ? [D: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
            = ( s @ B @ C ) )
       => ? [C: $i] :
          ! [D: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) ) ) )
            = ( s @ B @ D ) ) )
      & ( ? [C: $i] :
          ! [D: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) ) ) )
            = ( s @ B @ D ) )
       => ! [C: $i] :
          ? [D: $i] :
            ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ f ) @ ( s @ A @ D ) ) )
            = ( s @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3128]) ).

thf(651,axiom,
    ! [A: $i] :
      ( ( ! [B: $i,C: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) ) )
        & ! [B: $i] :
            ( ( ( s @ num @ B )
             != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_POSu_MONO) ).

thf(7071,plain,
    ! [A: $i] :
      ( ( ! [B: $i,C: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ C ) ) ) ) )
        & ! [B: $i] :
            ( ( ( s @ num @ B )
             != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ A ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[651]) ).

thf(2811,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ D ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ ( list @ A ) @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALL2u_ALL) ).

thf(19730,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) @ ( s @ A @ D ) ) ) @ ( s @ A @ D ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ A ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ ( list @ A ) @ D ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2811]) ).

thf(175,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_REFL) ).

thf(4066,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(2001,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) )
        = ( s @ ( fun @ A @ bool ) @ empty ) )
    <=> ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
         => ( ( s @ ( fun @ A @ bool ) @ C )
            = ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEMPTYu_UNIONS) ).

thf(15444,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) )
       => ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( ( s @ ( fun @ A @ bool ) @ C )
              = ( s @ ( fun @ A @ bool ) @ empty ) ) ) )
      & ( ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool ) ) @ in ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) ) )
           => ( ( s @ ( fun @ A @ bool ) @ C )
              = ( s @ ( fun @ A @ bool ) @ empty ) ) )
       => ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ A @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ B ) ) )
          = ( s @ ( fun @ A @ bool ) @ empty ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2001]) ).

thf(2133,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_ADDu_SYM) ).

thf(16110,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2133]) ).

thf(2802,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LEu_MIN) ).

thf(19689,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ A ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ C ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2802]) ).

thf(2099,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ D ) ) )
        = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ E ) ) ) )
    <=> ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) )
        & ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ D )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJPu_INJ) ).

thf(15894,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ D ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ E ) ) ) )
       => ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B )
            = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) )
          & ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ D )
            = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ E ) ) ) )
      & ( ( ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B )
            = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) )
          & ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ D )
            = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ E ) ) )
       => ( ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ B ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ D ) ) )
          = ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ num @ ( fun @ A @ bool ) ) @ ( fun @ num @ ( fun @ A @ bool ) ) ) ) @ injp ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ C ) ) ) @ ( s @ ( fun @ num @ ( fun @ A @ bool ) ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2099]) ).

thf(776,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ y ) ) ) @ ( s @ int @ x ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MINu_ACIu_conjunct0) ).

thf(7829,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ x ) ) ) @ ( s @ int @ y ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_min ) @ ( s @ int @ y ) ) ) @ ( s @ int @ x ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[776]) ).

thf(1116,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_1) ).

thf(9737,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1116]) ).

thf(1710,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_ADD) ).

thf(13574,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1710]) ).

thf(126,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_CMULu_RZERO) ).

thf(3817,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ r_r_ ) @ ( s @ real @ C ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ A ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(2837,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_INTER) ).

thf(19891,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2837]) ).

thf(2681,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) @ ( s @ real @ C ) ) ) )
        <=> ( ! [D: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
            & ! [D: $i] :
                ( ! [E: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ E ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ h_ ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ainf) ).

thf(19004,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) @ ( s @ real @ C ) ) ) )
           => ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
              & ! [D: $i] :
                  ( ! [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ E ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
          & ( ( ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ D ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
              & ! [D: $i] :
                  ( ! [E: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ E ) ) ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ E ) ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) )
     => ! [B: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ h_ ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ A ) @ ( s @ ( fun @ real @ bool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2681]) ).

thf(508,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ A @ E )
           != ( s @ A @ F ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ( ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ B @ D ) ) )
          = ( s @ B @ D ) )
        & ! [E: $i,F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ B @ D ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ B @ D ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ B @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_RECURSIONu_DELETE) ).

thf(6233,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ A @ E )
           != ( s @ A @ F ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ( ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ B @ D ) ) )
          = ( s @ B @ D ) )
        & ! [E: $i,F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ B @ D ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ B @ D ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ delete ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ B @ D ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).

thf(2475,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ oneu_one ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) )
    <=> ! [D: $i,E: $i] :
          ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
            = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
         => ( ( s @ B @ D )
            = ( s @ B @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aONEu_ONE) ).

thf(17927,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ oneu_one ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) )
       => ! [D: $i,E: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
           => ( ( s @ B @ D )
              = ( s @ B @ E ) ) ) )
      & ( ! [D: $i,E: $i] :
            ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ D ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ C ) @ ( s @ B @ E ) ) ) )
           => ( ( s @ B @ D )
              = ( s @ B @ E ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ bool ) @ oneu_one ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2475]) ).

thf(727,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',avectoru_sub) ).

thf(7493,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[727]) ).

thf(1380,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ! [G: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_LISTu_GEN) ).

thf(11462,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
                & ( ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) )
                & ( ! [G: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ bool ) ) @ mem ) @ ( s @ A @ G ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ bool @ ( fun @ ( list @ A ) @ bool ) ) ) @ setspec ) @ ( s @ ( list @ A ) @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ ( list @ A ) @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) )
     => ! [C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( list @ A ) @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1380]) ).

thf(338,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) )
    <=> ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_EQ) ).

thf(5236,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) )
       => ( ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( prod @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ B ) @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).

thf(427,axiom,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
      <=> ( ( s @ num @ A )
          = ( s @ num @ u_0 ) ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ A ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_PREu_conjunct2) ).

thf(5821,plain,
    ! [A: $i] :
    ? [B: $i] :
      ( ( ( p @ ( s @ bool @ B ) )
       => ( ( s @ num @ A )
          = ( s @ num @ u_0 ) ) )
      & ( ( ( s @ num @ A )
          = ( s @ num @ u_0 ) )
       => ( p @ ( s @ bool @ B ) ) )
      & ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ ( fun @ num @ num ) ) ) @ cond ) @ ( s @ bool @ B ) ) ) @ ( s @ num @ u_0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ pre ) @ ( s @ num @ A ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).

thf(2929,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( ( p @ ( s @ bool @ A ) )
         => ( p @ ( s @ bool @ B ) ) )
    <=> ( ( p @ ( s @ bool @ A ) )
        & ~ ( p @ ( s @ bool @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_IMP) ).

thf(20389,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( ( p @ ( s @ bool @ A ) )
           => ( p @ ( s @ bool @ B ) ) )
       => ( ( p @ ( s @ bool @ A ) )
          & ~ ( p @ ( s @ bool @ B ) ) ) )
      & ( ( ( p @ ( s @ bool @ A ) )
          & ~ ( p @ ( s @ bool @ B ) ) )
       => ~ ( ( p @ ( s @ bool @ A ) )
           => ( p @ ( s @ bool @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2929]) ).

thf(3061,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPSUBSETu_SUBSETu_TRANS) ).

thf(21148,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ psubset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3061]) ).

thf(215,axiom,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LINVu_LEMMA1) ).

thf(4395,plain,
    ! [A: $i,B: $i] :
      ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(1655,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_1u_conjunct2) ).

thf(13279,plain,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) ) )
     => ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1655]) ).

thf(1685,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_OFu_NUMu_SUB) ).

thf(13449,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
     => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ m_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1685]) ).

thf(2141,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ G ) ) ) @ ( s @ B @ F ) ) ) )
                 => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ G ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ B @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) ) ) )
         => ? [D: $i] :
            ! [E: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_REC) ).

thf(16142,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ l_l_ ) @ ( s @ B @ G ) ) ) @ ( s @ B @ F ) ) ) )
                 => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ G ) ) )
                    = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ G ) ) ) ) )
             => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ B @ F ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) ) ) )
         => ? [D: $i] :
            ! [E: $i] :
              ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ A ) ) @ C ) @ ( s @ ( fun @ B @ A ) @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2141]) ).

thf(906,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_SETu_OFu_LISTu_LE) ).

thf(8548,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ A @ bool ) ) @ setu_ofu_list ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( list @ A ) @ num ) @ length ) @ ( s @ ( list @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[906]) ).

thf(716,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ B @ F ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ) @ ( s @ ( list @ B ) @ F ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aALL2u_MAP) ).

thf(7430,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ B @ F ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ E ) @ ( s @ B @ F ) ) ) ) ) @ ( s @ B @ F ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ bool ) ) ) @ all2 ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( list @ B ) @ ( list @ A ) ) ) @ map0 ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ) @ ( s @ ( list @ B ) @ F ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ B ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( list @ B ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ B @ A ) @ E ) ) ) ) ) @ ( s @ ( list @ B ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[716]) ).

thf(1902,axiom,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINFINITEu_FROM) ).

thf(14796,plain,
    ! [A: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ bool ) @ infinite ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ from ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1902]) ).

thf(1270,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ m ) ) ) @ ( s @ int @ n ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ n ) ) ) @ ( s @ int @ m ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_ACu_conjunct0) ).

thf(10642,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ m ) ) ) @ ( s @ int @ n ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ n ) ) ) @ ( s @ int @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1270]) ).

thf(837,axiom,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ m ) ) ) @ ( s @ int @ n ) ) ) ) ) @ ( s @ int @ p1 ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ m ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ n ) ) ) @ ( s @ int @ p1 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_MULu_ACu_conjunct1) ).

thf(8189,plain,
    ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ m ) ) ) @ ( s @ int @ n ) ) ) ) ) @ ( s @ int @ p1 ) ) )
    = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ m ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ n ) ) ) @ ( s @ int @ p1 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[837]) ).

thf(104,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ real @ B )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ C )
        = ( s @ ( cart @ real @ A ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_MULu_LCANCELu_IMP) ).

thf(3740,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ real @ B )
         != ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
        & ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ C )
        = ( s @ ( cart @ real @ A ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(1730,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_MULu_LID) ).

thf(13678,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ ( cart @ real @ B ) @ B ) ) @ mat ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1730]) ).

thf(3140,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
    <=> ! [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUBSETu_PRED) ).

thf(21553,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
       => ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3140]) ).

thf(292,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) )
            & ! [E: $i,F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ G ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ F ) ) ) @ ( s @ real @ E ) ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ bool ) ) @ C ) @ ( s @ real @ E ) ) ) @ ( s @ A @ G ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ bool ) ) @ C ) @ ( s @ real @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) )
               => ? [F: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ bool ) ) @ C ) @ ( s @ real @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
         => ? [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
              & ! [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ bool ) ) @ C ) @ ( s @ real @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEPSILONu_DELTAu_MINIMAL) ).

thf(4955,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) ) )
          & ( ? [E: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ D ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ A @ E ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ) )
            & ! [E: $i,F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ G ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ F ) ) ) @ ( s @ real @ E ) ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ bool ) ) @ C ) @ ( s @ real @ E ) ) ) @ ( s @ A @ G ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ bool ) ) @ C ) @ ( s @ real @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) )
            & ! [E: $i] :
                ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ E ) ) ) )
               => ? [F: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ F ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ bool ) ) @ C ) @ ( s @ real @ F ) ) ) @ ( s @ A @ E ) ) ) ) ) ) )
         => ? [E: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ E ) ) ) )
              & ! [F: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ D ) @ ( s @ A @ F ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ A @ bool ) ) @ C ) @ ( s @ real @ E ) ) ) @ ( s @ A @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(177,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ C ) ) ) ) @ map2 ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ nil ) ) )
      = ( s @ ( list @ C ) @ nil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMAP2u_conjunct0) ).

thf(4071,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ ( list @ C ) @ ( i @ ( s @ ( fun @ ( list @ B ) @ ( list @ C ) ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ C ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( list @ A ) @ ( fun @ ( list @ B ) @ ( list @ C ) ) ) ) @ map2 ) @ ( s @ ( fun @ A @ ( fun @ B @ C ) ) @ f ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) @ ( s @ ( list @ B ) @ nil ) ) )
      = ( s @ ( list @ C ) @ nil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(616,axiom,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_INVu_0) ).

thf(6911,plain,
    p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).

thf(595,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ D ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) )
     => ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ div0 ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ C ) )
        & ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_DIVMODu_UNIQ) ).

thf(6778,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ C ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ D ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) )
     => ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ div0 ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ C ) )
        & ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ rem ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
          = ( s @ int @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).

thf(668,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
    <=> ? [B: $i] :
          ( ( ( s @ num @ B )
           != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ARCHu_INV) ).

thf(7176,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
       => ? [B: $i] :
            ( ( ( s @ num @ B )
             != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) )
      & ( ? [B: $i] :
            ( ( ( s @ num @ B )
             != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_inv ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[668]) ).

thf(964,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_LE) ).

thf(8876,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[964]) ).

thf(1561,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_ofu_num) ).

thf(12643,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ real @ int ) @ intu_ofu_real ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1561]) ).

thf(2431,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) )
          = ( s @ bool @ t ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) @ ( s @ ( fun @ A @ C ) @ F ) ) )
              = ( s @ ( fun @ B @ bool ) @ D ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) )
            <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asuperadmissible) ).

thf(17673,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ D ) @ ( s @ B @ E ) ) )
          = ( s @ bool @ t ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) @ ( s @ ( fun @ A @ C ) @ F ) ) )
              = ( s @ ( fun @ B @ bool ) @ D ) )
         => ! [F: $i,G: $i,H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) )
               => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) )
              & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) )
                 => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ tailadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ ( fun @ B @ A ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ B @ C ) ) @ I ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2431]) ).

thf(2003,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ F ) ) ) )
                 => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ A @ F ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) )
         => ( ? [D: $i] :
              ! [E: $i] :
                ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
            & ! [D: $i,E: $i] :
                ( ( ! [F: $i] :
                      ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                      = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ A @ F ) ) ) )
                  & ! [F: $i] :
                      ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) )
                      = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) )
               => ( ( s @ ( fun @ A @ B ) @ D )
                  = ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_EREC) ).

thf(15457,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
     => ! [C: $i] :
          ( ! [D: $i,E: $i,F: $i] :
              ( ! [G: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ G ) ) ) @ ( s @ A @ F ) ) ) )
                 => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ G ) ) )
                    = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
             => ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ A @ F ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) )
         => ( ? [D: $i] :
              ! [E: $i] :
                ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ E ) ) )
                = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
            & ! [D: $i,E: $i] :
                ( ( ! [F: $i] :
                      ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ D ) @ ( s @ A @ F ) ) )
                      = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) @ ( s @ A @ F ) ) ) )
                  & ! [F: $i] :
                      ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ F ) ) )
                      = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ C ) @ ( s @ ( fun @ A @ B ) @ E ) ) ) @ ( s @ A @ F ) ) ) ) )
               => ( ( s @ ( fun @ A @ B ) @ D )
                  = ( s @ ( fun @ A @ B ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2003]) ).

thf(687,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ A @ E )
           != ( s @ A @ F ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ( ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ B @ D ) ) )
          = ( s @ B @ D ) )
        & ! [E: $i,F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ B @ D ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ B @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ B @ D ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_RECURSION) ).

thf(7252,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i,G: $i] :
          ( ( ( s @ A @ E )
           != ( s @ A @ F ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ F ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ G ) ) ) ) ) ) )
     => ( ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ empty ) ) ) @ ( s @ B @ D ) ) )
          = ( s @ B @ D ) )
        & ! [E: $i,F: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ B @ D ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ B @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) @ ( s @ A @ E ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ B @ D ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[687]) ).

thf(1291,axiom,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ u_0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aARITHu_ZEROu_conjunct1) ).

thf(10758,plain,
    ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ u_0 ) ) )
    = ( s @ num @ u_0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1291]) ).

thf(2955,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ empty ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPu_SING) ).

thf(20531,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ sup ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) ) @ insert ) @ ( s @ real @ A ) ) ) @ ( s @ ( fun @ real @ bool ) @ empty ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2955]) ).

thf(2459,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ? [D: $i] : ( p @ ( s @ bool @ B ) )
        & ? [D: $i] : ( p @ ( s @ bool @ C ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ B ) )
          & ( p @ ( s @ bool @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTRIVu_ANDu_EXISTSu_THM) ).

thf(17849,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2459]) ).

thf(145,axiom,
    ! [A: $i] :
      ( ( ( s @ num @ A )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
      | ? [B: $i] :
          ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_CASES) ).

thf(3918,plain,
    ! [A: $i] :
      ( ( ( s @ num @ A )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
      | ? [B: $i] :
          ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ suc ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(3086,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_ADD) ).

thf(21272,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ B ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3086]) ).

thf(3169,axiom,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_REFL) ).

thf(21693,plain,
    ! [A: $i] :
      ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3169]) ).

thf(1999,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCOUNTABLEu_INTER) ).

thf(15429,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ countable ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ inter ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1999]) ).

thf(1447,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i,C: $i] :
        ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LINVu_LEMMA6) ).

thf(11925,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i,C: $i] :
        ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ E ) ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ E ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1447]) ).

thf(2735,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) )
    <=> ? [C: $i] :
          ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aintu_divides) ).

thf(19298,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) )
       => ? [C: $i] :
            ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      & ( ? [C: $i] :
            ( ( s @ int @ A )
            = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_divides ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2735]) ).

thf(1578,axiom,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) )
      & ! [E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ E ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ E ) ) ) @ ( s @ num @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_INDEXu_INRANGE) ).

thf(12717,plain,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) )
      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ B @ bool ) @ univ ) ) ) ) ) )
      & ! [E: $i] :
          ( ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ E ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ A @ ( i @ ( s @ ( fun @ num @ A ) @ ( i @ ( s @ ( fun @ ( cart @ A @ B ) @ ( fun @ num @ A ) ) @ d_ ) @ ( s @ ( cart @ A @ B ) @ E ) ) ) @ ( s @ num @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1578]) ).

thf(1179,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_LTu_ADDL) ).

thf(10060,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_add ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1179]) ).

thf(1185,axiom,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( list @ A ) @ nil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREVERSEu_conjunct0) ).

thf(10086,plain,
    ! [A: $i] :
      ( ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ reverse ) @ ( s @ ( list @ A ) @ nil ) ) )
      = ( s @ ( list @ A ) @ nil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1185]) ).

thf(2339,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ delete ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) @ ( s @ B @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_DELETE) ).

thf(17123,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ B @ bool ) ) ) @ delete ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ B @ E ) ) ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ B @ ( cart @ real @ A ) ) @ C ) @ ( s @ B @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2339]) ).

thf(2995,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ? [I: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) )
                    = ( s @ B @ H ) ) )
              & ! [I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) )
                      = ( s @ B @ H ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ J ) ) )
                      = ( s @ B @ H ) ) )
                 => ( ( s @ A @ I )
                    = ( s @ A @ J ) ) ) ) )
        & ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ H ) ) ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ E ) @ ( s @ A @ H ) ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_EQu_GENERAL) ).

thf(20744,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
           => ( ? [I: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) )
                    = ( s @ B @ H ) ) )
              & ! [I: $i,J: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ I ) ) )
                      = ( s @ B @ H ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ J ) ) )
                      = ( s @ B @ H ) ) )
                 => ( ( s @ A @ I )
                    = ( s @ A @ J ) ) ) ) )
        & ! [H: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ H ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
              & ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ G ) @ ( s @ A @ H ) ) ) ) )
                = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ E ) @ ( s @ A @ H ) ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ E ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ num ) @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2995]) ).

thf(28,axiom,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
      = ( s @ A @ h0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHD) ).

thf(3296,plain,
    ! [A: $i] :
      ( ( s @ A @ ( i @ ( s @ ( fun @ ( list @ A ) @ A ) @ hd ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
      = ( s @ A @ h0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(239,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_CLAUSESu_conjunct3) ).

thf(4533,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ num @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(2515,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ADDu_RDISTRIB) ).

thf(18139,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) @ ( s @ real @ C ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2515]) ).

thf(156,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ A ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ univ ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_FINITEu_IMAGE) ).

thf(3960,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ ( finite_image @ A ) @ bool ) @ num ) @ card ) @ ( s @ ( fun @ ( finite_image @ A ) @ bool ) @ univ ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ dimindex ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(1231,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ r0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ r0 ) @ ( s @ A @ h0 ) ) ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ r0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aPAIRWISEu_conjunct1) ).

thf(10389,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ r0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ r0 ) @ ( s @ A @ h0 ) ) ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ r0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ r0 ) @ ( s @ A @ h0 ) ) ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ r0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( list @ A ) @ bool ) ) @ pairwise0 ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ r0 ) ) ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ h0 ) ) ) @ ( s @ ( list @ A ) @ t0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1231]) ).

thf(1229,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_OFu_NUMu_LE) ).

thf(10383,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1229]) ).

thf(2243,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_ADDu_SYM) ).

thf(16679,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_add ) @ ( s @ ( prod @ hreal @ hreal ) @ B ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2243]) ).

thf(1131,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_PAIR) ).

thf(9838,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ B ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) )
     => ! [B: $i,C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ B ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( fun @ num @ real ) ) @ A ) @ ( s @ ( fun @ num @ real ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1131]) ).

thf(937,axiom,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_ABSu_NUM) ).

thf(8786,plain,
    ! [A: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[937]) ).

thf(732,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i] :
          ( ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) )
          = ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anaddu_ofu_num) ).

thf(7548,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
     => ! [B: $i] :
          ( ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ B ) ) )
          = ( s @ nadd @ ( i @ ( s @ ( fun @ ( fun @ num @ num ) @ nadd ) @ mku_nadd ) @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[732]) ).

thf(249,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVSUMu_1) ).

thf(4655,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ ( fun @ num @ ( cart @ real @ A ) ) @ ( cart @ real @ A ) ) ) @ vsum ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( fun @ num @ bool ) ) ) @ o_o_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ f ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(2721,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_HASu_SIZEu_CONG) ).

thf(19232,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ num @ E ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ a_u_c ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2721]) ).

thf(73,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ? [E: $i,F: $i] :
          ( ! [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
                  = ( s @ A @ G ) ) ) )
          & ! [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) )
                  = ( s @ B @ G ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_EQu_BIJECTIONS) ).

thf(3580,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
        & ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ C ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) )
     => ? [E: $i,F: $i] :
          ( ! [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ G ) ) ) ) )
                  = ( s @ A @ G ) ) ) )
          & ! [G: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
             => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ E ) @ ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ F ) @ ( s @ B @ G ) ) ) ) )
                  = ( s @ B @ G ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(2728,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aBILINEARu_RSUB) ).

thf(19279,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ bool ) @ bilinear ) @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) ) ) )
     => ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ C ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) )
        = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ F ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ C ) @ ( cart @ real @ A ) ) ) @ D ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ ( cart @ real @ C ) @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2728]) ).

thf(396,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
        <=> ? [G: $i,H: $i] :
              ( ( ( p @ ( s @ bool @ H ) )
              <=> ? [I: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
          = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONSu_IMAGE) ).

thf(5591,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) )
           => ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) ) )
          & ( ? [G: $i,H: $i] :
                ( ( ( p @ ( s @ bool @ H ) )
                 => ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) ) )
                & ( ? [I: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ I ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ I ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ H ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ B @ G ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ B @ F ) ) ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ B @ bool ) ) @ unions ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
          = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ bool ) ) ) @ C ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).

thf(2916,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_LEu_WELLDEFu_LEMMA) ).

thf(20301,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ C ) ) ) @ ( s @ nadd @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_le ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2916]) ).

thf(1284,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_LID) ).

thf(10723,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ A ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1284]) ).

thf(2113,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ B ) ) ) @ ( s @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_REFL) ).

thf(15985,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ B ) ) ) @ ( s @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2113]) ).

thf(1307,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ F ) ) ) )
        <=> ! [G: $i,H: $i] :
            ? [I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ? [J: $i,K: $i,L: $i] :
                    ( ! [M: $i,N: $i,O: $i] :
                        ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) )
                          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ O ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) )
                       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ O ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ N ) ) ) ) ) ) )
                    & ! [M: $i,N: $i,O: $i] :
                        ( ! [P: $i] :
                            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ P ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ O ) ) ) ) ) )
                           => ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ P ) ) )
                              = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ N ) @ ( s @ C @ P ) ) ) ) )
                       => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                            = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) )
                          & ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                            = ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) )
                          & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                            = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) ) ) )
                    & ! [M: $i,N: $i] :
                        ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) )
                        = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ C ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
     => ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ F ) ) ) ) )
         => ? [G: $i] :
            ! [H: $i] :
              ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ G ) @ ( s @ C @ H ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ F ) ) ) @ ( s @ ( fun @ C @ B ) @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_RECu_CASES) ).

thf(10813,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ F ) ) ) )
           => ! [G: $i,H: $i] :
              ? [I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ? [J: $i,K: $i,L: $i] :
                      ( ! [M: $i,N: $i,O: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ O ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ O ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ N ) ) ) ) ) ) )
                      & ! [M: $i,N: $i,O: $i] :
                          ( ! [P: $i] :
                              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ P ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ O ) ) ) ) ) )
                             => ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ P ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ N ) @ ( s @ C @ P ) ) ) ) )
                         => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) )
                            & ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) )
                            & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) ) ) )
                      & ! [M: $i,N: $i] :
                          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) )
                          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) ) )
                & ( ? [J: $i,K: $i,L: $i] :
                      ( ! [M: $i,N: $i,O: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ O ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ O ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ N ) ) ) ) ) ) )
                      & ! [M: $i,N: $i,O: $i] :
                          ( ! [P: $i] :
                              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ P ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ O ) ) ) ) ) )
                             => ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ P ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ N ) @ ( s @ C @ P ) ) ) ) )
                         => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) )
                            & ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) )
                            & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) ) ) )
                      & ! [M: $i,N: $i] :
                          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) )
                          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ C ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) ) )
          & ( ! [G: $i,H: $i] :
              ? [I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ? [J: $i,K: $i,L: $i] :
                      ( ! [M: $i,N: $i,O: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ O ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ O ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ N ) ) ) ) ) ) )
                      & ! [M: $i,N: $i,O: $i] :
                          ( ! [P: $i] :
                              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ P ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ O ) ) ) ) ) )
                             => ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ P ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ N ) @ ( s @ C @ P ) ) ) ) )
                         => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) )
                            & ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) )
                            & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) ) ) )
                      & ! [M: $i,N: $i] :
                          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) )
                          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) ) )
                & ( ? [J: $i,K: $i,L: $i] :
                      ( ! [M: $i,N: $i,O: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ O ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) )
                         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ O ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ N ) ) ) ) ) ) )
                      & ! [M: $i,N: $i,O: $i] :
                          ( ! [P: $i] :
                              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ C @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) @ ( s @ C @ P ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ G ) @ ( s @ A @ O ) ) ) ) ) )
                             => ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ P ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ N ) @ ( s @ C @ P ) ) ) ) )
                         => ( ( ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) )
                            & ( ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) )
                            & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ O ) ) )
                              = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ N ) ) ) @ ( s @ A @ O ) ) ) ) ) )
                      & ! [M: $i,N: $i] :
                          ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ H ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) )
                          = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ ( fun @ B @ B ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ bool ) ) @ J ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ M ) @ ( s @ C @ ( i @ ( s @ ( fun @ A @ C ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ C ) ) @ K ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ L ) @ ( s @ ( fun @ C @ B ) @ M ) ) ) @ ( s @ A @ N ) ) ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ geq ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ F ) @ ( s @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ C ) @ ( fun @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) ) @ c_ ) @ ( s @ ( fun @ A @ C ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) @ H ) ) ) ) ) ) ) @ ( s @ bool @ I ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) ) ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ F ) ) ) ) ) )
     => ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) ) @ gabs ) @ ( s @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ C @ bool ) ) @ ( fun @ ( fun @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ C @ ( fun @ C @ bool ) ) @ E ) ) ) ) ) ) ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ F ) ) ) ) )
         => ? [G: $i] :
            ! [H: $i] :
              ( ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ G ) @ ( s @ C @ H ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ C @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ B ) ) @ ( i @ ( s @ ( fun @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ C @ B ) ) ) @ casewise ) @ ( s @ ( list @ ( prod @ ( fun @ A @ C ) @ ( fun @ ( fun @ C @ B ) @ ( fun @ A @ B ) ) ) ) @ F ) ) ) @ ( s @ ( fun @ C @ B ) @ G ) ) ) @ ( s @ C @ H ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1307]) ).

thf(384,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ ( cart @ real @ A ) @ B )
       != ( s @ ( cart @ real @ A ) @ C ) )
    <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_NZ) ).

thf(5542,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( s @ ( cart @ real @ A ) @ B )
         != ( s @ ( cart @ real @ A ) @ C ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) ) )
      & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) )
       => ( ( s @ ( cart @ real @ A ) @ B )
         != ( s @ ( cart @ real @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).

thf(2999,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
        <=> ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
        <=> ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNOTu_ALL) ).

thf(20776,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) )
           => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) ) )
          & ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ C ) @ ( s @ A @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) )
          & ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ ex0 ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) )
           => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( list @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( list @ A ) @ bool ) ) @ all0 ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( list @ A ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2999]) ).

thf(2429,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SGNu_MUL) ).

thf(17661,plain,
    ! [A: $i,B: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_sgn ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2429]) ).

thf(2830,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ realu_mod ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
    <=> ? [D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ D ) ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_mod) ).

thf(19856,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ realu_mod ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
       => ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ D ) ) ) )
            & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) ) )
      & ( ? [D: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ integer ) @ ( s @ real @ D ) ) ) )
            & ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) )
              = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ D ) ) ) @ ( s @ real @ C ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ realu_mod ) @ ( s @ real @ C ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2830]) ).

thf(353,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ? [H: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ H ) ) ) @ ( s @ A @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ E ) ) ) @ ( s @ A @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ E ) ) ) @ ( s @ A @ G ) ) ) ) )
             => ( ( s @ A @ F )
                = ( s @ A @ G ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_RELATIONAL) ).

thf(5307,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ? [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ H ) ) ) @ ( s @ A @ F ) ) ) ) ) )
                & ( ? [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ H ) ) ) @ ( s @ A @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ? [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ H ) ) ) @ ( s @ A @ F ) ) ) ) ) )
                & ( ? [H: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ H ) ) ) @ ( s @ A @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ E ) ) ) @ ( s @ A @ F ) ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) @ ( s @ B @ E ) ) ) @ ( s @ A @ G ) ) ) ) )
             => ( ( s @ A @ F )
                = ( s @ A @ G ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) ) @ C ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ D ) ) ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ s0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).

thf(2178,axiom,
    ! [A: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aEVENu_ANDu_ODD) ).

thf(16346,plain,
    ! [A: $i] :
      ~ ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ even ) @ ( s @ num @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2178]) ).

thf(2400,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_ADDu_LCANCEL) ).

thf(17497,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ B ) ) ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ nadd @ nadd ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ nadd ) ) @ naddu_add ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ C ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ B ) ) ) @ ( s @ nadd @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2400]) ).

thf(3113,axiom,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ bool @ A ) @ D ) @ ( s @ bool @ f0 ) ) )
        = ( s @ A @ B ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ bool @ A ) @ D ) @ ( s @ bool @ t ) ) )
        = ( s @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aboolu_RECURSION) ).

thf(21429,plain,
    ! [A: $i,B: $i,C: $i] :
    ? [D: $i] :
      ( ( ( s @ A @ ( i @ ( s @ ( fun @ bool @ A ) @ D ) @ ( s @ bool @ f0 ) ) )
        = ( s @ A @ B ) )
      & ( ( s @ A @ ( i @ ( s @ ( fun @ bool @ A ) @ D ) @ ( s @ bool @ t ) ) )
        = ( s @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3113]) ).

thf(1719,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',avectoru_add) ).

thf(13631,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ num @ E ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ d_ ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) @ ( s @ num @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_add ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( fun @ num @ real ) ) ) @ B ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1719]) ).

thf(1171,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ G ) ) ) ) )
         => ( ( s @ A @ F )
            = ( s @ A @ G ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ D ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ num ) ) ) @ o ) @ ( s @ ( fun @ B @ num ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_IMAGE) ).

thf(10013,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ F ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ A @ B ) @ C ) @ ( s @ A @ G ) ) ) ) )
         => ( ( s @ A @ F )
            = ( s @ A @ G ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) @ ( s @ ( fun @ B @ num ) @ D ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ num ) ) ) @ o ) @ ( s @ ( fun @ B @ num ) @ D ) ) ) @ ( s @ ( fun @ A @ B ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1171]) ).

thf(2844,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
     => ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) )
      <=> ( ( s @ real @ B )
          = ( s @ real @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_POWu_EQu_ODDu_EQ) ).

thf(19923,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ odd ) @ ( s @ num @ A ) ) ) )
     => ( ( ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) )
         => ( ( s @ real @ B )
            = ( s @ real @ C ) ) )
        & ( ( ( s @ real @ B )
            = ( s @ real @ C ) )
         => ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ B ) ) ) @ ( s @ num @ A ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ C ) ) ) @ ( s @ num @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2844]) ).

thf(980,axiom,
    ! [A: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIVu_REFL) ).

thf(8991,plain,
    ! [A: $i] :
      ( ( ( s @ num @ A )
       != ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ A ) ) ) @ ( s @ num @ A ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[980]) ).

thf(29,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anaddu_rinv) ).

thf(3299,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ div ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ destu_nadd ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(2584,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ F ) ) ) )
            & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) )
         => ( ( s @ B @ E )
            = ( s @ B @ F ) ) )
    <=> ! [E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ F ) ) ) ) )
         => ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) )
              = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) )
          <=> ( ( s @ B @ E )
              = ( s @ B @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINJECTIVEu_ONu_ALT) ).

thf(18510,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ F ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) )
           => ( ( s @ B @ E )
              = ( s @ B @ F ) ) )
       => ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ F ) ) ) ) )
           => ( ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) )
               => ( ( s @ B @ E )
                  = ( s @ B @ F ) ) )
              & ( ( ( s @ B @ E )
                  = ( s @ B @ F ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) ) ) ) )
      & ( ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ F ) ) ) ) )
           => ( ( ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) )
               => ( ( s @ B @ E )
                  = ( s @ B @ F ) ) )
              & ( ( ( s @ B @ E )
                  = ( s @ B @ F ) )
               => ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) )
                  = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) ) ) )
       => ! [E: $i,F: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ E ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ C ) @ ( s @ B @ F ) ) ) )
              & ( ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ E ) ) )
                = ( s @ A @ ( i @ ( s @ ( fun @ B @ A ) @ D ) @ ( s @ B @ F ) ) ) ) )
           => ( ( s @ B @ E )
              = ( s @ B @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2584]) ).

thf(3154,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_ABS) ).

thf(21613,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) @ ( s @ A @ D ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ D ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
         => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ ( fun @ A @ real ) ) @ B ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3154]) ).

thf(267,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ g ) @ ( s @ A @ E ) ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ g ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_EQu_SUPERSET) ).

thf(4723,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [E: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ g ) @ ( s @ A @ E ) ) ) ) )
        & ! [E: $i] :
            ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
              & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
           => ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ B ) @ ( s @ A @ E ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
     => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ B ) ) )
        = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ num ) @ g ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(2461,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_UNIONu_IMP) ).

thf(17861,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2461]) ).

thf(1970,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i,I: $i] :
          ( ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ D ) @ H ) ) ) @ ( s @ B @ I ) ) )
          = ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ u_seqpattern ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ D ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ A @ D ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i] :
              ( ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ A @ D ) @ I ) ) )
              = ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ A @ D ) @ I ) ) ) )
         => ! [G: $i] :
              ( ! [H: $i,I: $i,J: $i,K: $i,L: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ J ) ) ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) )
                <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ H ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) )
                    & ~ ? [M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) ) ) @ ( s @ D @ M ) ) ) ) ) )
             => ! [H: $i] :
                  ( ! [I: $i,J: $i,K: $i,L: $i] :
                      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ K ) ) ) @ ( s @ ( fun @ A @ D ) @ L ) ) )
                      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ K ) ) ) @ ( s @ ( fun @ A @ D ) @ L ) ) ) )
                 => ! [I: $i] :
                      ( ! [J: $i,K: $i,L: $i,M: $i] :
                          ( ( s @ D @ ( i @ ( s @ ( fun @ B @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ A @ D ) @ L ) ) ) @ ( s @ B @ M ) ) )
                          = ( s @ D @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ D ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ D ) ) @ u_match ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ D ) @ L ) ) ) @ ( s @ B @ M ) ) ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ A @ D ) @ L ) ) ) @ ( s @ B @ M ) ) ) ) ) )
                     => ! [J: $i] :
                          ( ! [K: $i,L: $i,M: $i] :
                              ( ( s @ ( fun @ B @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ A @ D ) @ M ) ) )
                              = ( s @ ( fun @ B @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ A @ D ) @ M ) ) ) )
                         => ! [K: $i] :
                              ( ! [L: $i,M: $i,N: $i,O: $i,P: $i] :
                                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ N ) ) ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) )
                                <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ L ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) )
                                    & ? [Q: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ N ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) ) ) @ ( s @ D @ Q ) ) ) ) ) )
                             => ! [L: $i] :
                                  ( ! [M: $i,N: $i,O: $i,P: $i] :
                                      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ O ) ) ) @ ( s @ ( fun @ A @ D ) @ P ) ) )
                                      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ O ) ) ) @ ( s @ ( fun @ A @ D ) @ P ) ) ) )
                                 => ! [M: $i] :
                                      ( ! [N: $i,O: $i,P: $i,Q: $i] :
                                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ O ) ) ) @ ( s @ ( fun @ A @ D ) @ P ) ) ) @ ( s @ B @ Q ) ) ) )
                                        <=> ? [R: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ N ) @ ( s @ ( fun @ A @ D ) @ P ) ) ) @ ( s @ B @ Q ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ O ) @ ( s @ ( fun @ A @ D ) @ P ) ) ) @ ( s @ B @ Q ) ) ) ) ) @ ( s @ D @ R ) ) ) ) )
                                     => ! [N: $i] :
                                          ( ! [O: $i,P: $i,Q: $i] :
                                              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ P ) ) ) @ ( s @ ( fun @ A @ D ) @ Q ) ) )
                                              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ P ) ) ) @ ( s @ ( fun @ A @ D ) @ Q ) ) ) )
                                         => ! [O: $i,P: $i,Q: $i,R: $i,S: $i,T: $i] :
                                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ B @ A ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) ) ) @ ( s @ ( fun @ B @ A ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ R ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) ) ) @ ( s @ ( fun @ B @ A ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ S ) ) ) ) ) ) )
                                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ B @ A ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPERADMISSIBLEu_MATCHu_SEQPATTERN) ).

thf(15219,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i,G: $i,H: $i,I: $i] :
          ( ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ F ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ ( fun @ A @ D ) @ H ) ) ) @ ( s @ B @ I ) ) )
          = ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ u_seqpattern ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ F ) @ ( s @ ( fun @ A @ D ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) @ ( s @ ( fun @ A @ D ) @ H ) ) ) @ ( s @ B @ I ) ) ) ) ) )
     => ! [F: $i] :
          ( ! [G: $i,H: $i,I: $i] :
              ( ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ A @ D ) @ I ) ) )
              = ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) ) @ E ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ G ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ H ) ) ) @ ( s @ ( fun @ A @ D ) @ I ) ) ) )
         => ! [G: $i] :
              ( ! [H: $i,I: $i,J: $i,K: $i,L: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ J ) ) ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) )
                   => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ H ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) )
                      & ~ ? [M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) ) ) @ ( s @ D @ M ) ) ) ) ) )
                  & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ H ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) )
                      & ~ ? [M: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) ) ) @ ( s @ D @ M ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ J ) ) ) @ ( s @ ( fun @ A @ D ) @ K ) ) ) @ ( s @ B @ L ) ) ) ) ) )
             => ! [H: $i] :
                  ( ! [I: $i,J: $i,K: $i,L: $i] :
                      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ K ) ) ) @ ( s @ ( fun @ A @ D ) @ L ) ) )
                      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ K ) ) ) @ ( s @ ( fun @ A @ D ) @ L ) ) ) )
                 => ! [I: $i] :
                      ( ! [J: $i,K: $i,L: $i,M: $i] :
                          ( ( s @ D @ ( i @ ( s @ ( fun @ B @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ A @ D ) @ L ) ) ) @ ( s @ B @ M ) ) )
                          = ( s @ D @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ D ) @ ( i @ ( s @ ( fun @ C @ ( fun @ ( fun @ C @ ( fun @ D @ bool ) ) @ D ) ) @ u_match ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ J ) @ ( s @ ( fun @ A @ D ) @ L ) ) ) @ ( s @ B @ M ) ) ) ) ) @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ K ) @ ( s @ ( fun @ A @ D ) @ L ) ) ) @ ( s @ B @ M ) ) ) ) ) )
                     => ! [J: $i] :
                          ( ! [K: $i,L: $i,M: $i] :
                              ( ( s @ ( fun @ B @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ A @ D ) @ M ) ) )
                              = ( s @ ( fun @ B @ D ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ L ) ) ) @ ( s @ ( fun @ A @ D ) @ M ) ) ) )
                         => ! [K: $i] :
                              ( ! [L: $i,M: $i,N: $i,O: $i,P: $i] :
                                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ N ) ) ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) )
                                   => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ L ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) )
                                      & ? [Q: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ N ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) ) ) @ ( s @ D @ Q ) ) ) ) ) )
                                  & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ L ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) )
                                      & ? [Q: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ N ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) ) ) @ ( s @ D @ Q ) ) ) ) )
                                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ N ) ) ) @ ( s @ ( fun @ A @ D ) @ O ) ) ) @ ( s @ B @ P ) ) ) ) ) )
                             => ! [L: $i] :
                                  ( ! [M: $i,N: $i,O: $i,P: $i] :
                                      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ O ) ) ) @ ( s @ ( fun @ A @ D ) @ P ) ) )
                                      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ O ) ) ) @ ( s @ ( fun @ A @ D ) @ P ) ) ) )
                                 => ! [M: $i] :
                                      ( ! [N: $i,O: $i,P: $i,Q: $i] :
                                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ O ) ) ) @ ( s @ ( fun @ A @ D ) @ P ) ) ) @ ( s @ B @ Q ) ) ) )
                                           => ? [R: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ N ) @ ( s @ ( fun @ A @ D ) @ P ) ) ) @ ( s @ B @ Q ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ O ) @ ( s @ ( fun @ A @ D ) @ P ) ) ) @ ( s @ B @ Q ) ) ) ) ) @ ( s @ D @ R ) ) ) ) )
                                          & ( ? [R: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ D @ bool ) @ ( i @ ( s @ ( fun @ C @ ( fun @ D @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ N ) @ ( s @ ( fun @ A @ D ) @ P ) ) ) @ ( s @ B @ Q ) ) ) @ ( s @ C @ ( i @ ( s @ ( fun @ B @ C ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ O ) @ ( s @ ( fun @ A @ D ) @ P ) ) ) @ ( s @ B @ Q ) ) ) ) ) @ ( s @ D @ R ) ) ) )
                                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ O ) ) ) @ ( s @ ( fun @ A @ D ) @ P ) ) ) @ ( s @ B @ Q ) ) ) ) ) )
                                     => ! [N: $i] :
                                          ( ! [O: $i,P: $i,Q: $i] :
                                              ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ P ) ) ) @ ( s @ ( fun @ A @ D ) @ Q ) ) )
                                              = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ M ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ P ) ) ) @ ( s @ ( fun @ A @ D ) @ Q ) ) ) )
                                         => ! [O: $i,P: $i,Q: $i,R: $i,S: $i,T: $i] :
                                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ B @ A ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ N ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) ) ) @ ( s @ ( fun @ B @ A ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ R ) ) ) ) ) )
                                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) ) ) @ ( s @ ( fun @ B @ A ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ S ) ) ) ) ) ) )
                                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ bool ) ) ) ) @ superadmissible ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ B @ A ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ D ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ C ) ) @ T ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) ) ) @ F ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ A @ D ) @ ( fun @ B @ ( fun @ C @ ( fun @ D @ bool ) ) ) ) @ S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1970]) ).

thf(695,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITERATEu_DIFF) ).

thf(7310,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ bool ) @ monoidal ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) )
     => ! [D: $i,E: $i,F: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) )
         => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) )
            = ( s @ B @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ B ) @ B ) ) ) @ iterate ) @ ( s @ ( fun @ B @ ( fun @ B @ B ) ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[695]) ).

thf(39,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arealu_abs) ).

thf(3405,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ A ) ) )
      = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(1907,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_MULu_DOT) ).

thf(14811,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) @ ( s @ num @ F ) ) )
          = ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ num @ ( cart @ real @ B ) ) ) @ d_ ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ num @ F ) ) ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) )
     => ! [D: $i,E: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( fun @ num @ real ) ) ) @ C ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1907]) ).

thf(1695,axiom,
    ! [A: $i] :
      ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ B ) ) ) )
    <=> ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n10 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFORALLu_VECTORu_1) ).

thf(13493,plain,
    ! [A: $i] :
      ( ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ B ) ) ) )
       => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n10 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) ) )
      & ( ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( cart @ A @ n10 ) ) @ vector ) @ ( s @ ( list @ A ) @ ( i @ ( s @ ( fun @ ( list @ A ) @ ( list @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( list @ A ) @ ( list @ A ) ) ) @ cons ) @ ( s @ A @ B ) ) ) @ ( s @ ( list @ A ) @ nil ) ) ) ) ) ) ) )
       => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ A @ n10 ) @ bool ) @ p0 ) @ ( s @ ( cart @ A @ n10 ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1695]) ).

thf(2688,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ D ) ) )
        = ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ G ) ) ) )
    <=> ( ( ( s @ num @ B )
          = ( s @ num @ E ) )
        & ( ( s @ A @ C )
          = ( s @ A @ F ) )
        & ( ( s @ ( fun @ num @ ( recspace @ A ) ) @ D )
          = ( s @ ( fun @ num @ ( recspace @ A ) ) @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCONSTRu_INJ) ).

thf(19072,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ D ) ) )
          = ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ G ) ) ) )
       => ( ( ( s @ num @ B )
            = ( s @ num @ E ) )
          & ( ( s @ A @ C )
            = ( s @ A @ F ) )
          & ( ( s @ ( fun @ num @ ( recspace @ A ) ) @ D )
            = ( s @ ( fun @ num @ ( recspace @ A ) ) @ G ) ) ) )
      & ( ( ( ( s @ num @ B )
            = ( s @ num @ E ) )
          & ( ( s @ A @ C )
            = ( s @ A @ F ) )
          & ( ( s @ ( fun @ num @ ( recspace @ A ) ) @ D )
            = ( s @ ( fun @ num @ ( recspace @ A ) ) @ G ) ) )
       => ( ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ B ) ) ) @ ( s @ A @ C ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ D ) ) )
          = ( s @ ( recspace @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ A @ ( fun @ ( fun @ num @ ( recspace @ A ) ) @ ( recspace @ A ) ) ) ) @ constr ) @ ( s @ num @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ num @ ( recspace @ A ) ) @ G ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2688]) ).

thf(2608,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) @ ( s @ A @ G ) ) ) ) )
        & ! [G: $i,H: $i,I: $i] :
            ( ( ( s @ A @ G )
             != ( s @ A @ H ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) @ ( s @ A @ G ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) @ ( s @ A @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) ) )
        & ! [G: $i,H: $i,I: $i] :
            ( ( ( s @ A @ G )
             != ( s @ A @ H ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) @ ( s @ A @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) )
     => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ B @ F ) ) )
        = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ B @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aITSETu_EQ) ).

thf(18675,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
        & ! [G: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
           => ( ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) @ ( s @ A @ G ) ) )
              = ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) @ ( s @ A @ G ) ) ) ) )
        & ! [G: $i,H: $i,I: $i] :
            ( ( ( s @ A @ G )
             != ( s @ A @ H ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) @ ( s @ A @ G ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) @ ( s @ A @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) ) )
        & ! [G: $i,H: $i,I: $i] :
            ( ( ( s @ A @ G )
             != ( s @ A @ H ) )
           => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) @ ( s @ A @ H ) ) ) @ ( s @ B @ I ) ) ) ) )
              = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) @ ( s @ A @ H ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) @ ( s @ A @ G ) ) ) @ ( s @ B @ I ) ) ) ) ) ) ) )
     => ( ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ D ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ B @ F ) ) )
        = ( s @ B @ ( i @ ( s @ ( fun @ B @ B ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ B ) ) ) @ itset ) @ ( s @ ( fun @ A @ ( fun @ B @ B ) ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ B @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2608]) ).

thf(345,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_ADD) ).

thf(5261,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) @ ( s @ A @ E ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ C ) @ ( s @ A @ E ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ D ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
         => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ ( fun @ ( fun @ A @ num ) @ ( fun @ A @ num ) ) ) @ B ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).

thf(835,axiom,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSQRTu_MONOu_LEu_EQ) ).

thf(8179,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ A ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ A ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ sqrt ) @ ( s @ real @ B ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[835]) ).

thf(367,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) )
        <=> ? [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ F ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_REALu_INTERVALu_conjunct6) ).

thf(5436,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i,D: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) )
           => ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) ) )
          & ( ? [E: $i,F: $i] :
                ( ( ( p @ ( s @ bool @ F ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ E ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ E ) ) ) @ ( s @ real @ C ) ) ) ) )
                 => ( p @ ( s @ bool @ F ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ bool @ ( fun @ real @ bool ) ) ) @ setspec ) @ ( s @ real @ D ) ) ) @ ( s @ bool @ F ) ) ) @ ( s @ real @ E ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) ) )
     => ! [B: $i,C: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ ( fun @ real @ bool ) ) @ gspec ) @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ ( fun @ real @ bool ) ) ) @ A ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).

thf(1348,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
          <=> ( ( s @ num @ D )
              = ( s @ num @ C ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) @ ( s @ num @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',abasis) ).

thf(11241,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i] :
        ? [E: $i] :
          ( ( ( p @ ( s @ bool @ E ) )
           => ( ( s @ num @ D )
              = ( s @ num @ C ) ) )
          & ( ( ( s @ num @ D )
              = ( s @ num @ C ) )
           => ( p @ ( s @ bool @ E ) ) )
          & ( ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) )
            = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ E ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
     => ! [C: $i] :
          ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) )
          = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( fun @ num @ real ) @ ( cart @ real @ A ) ) @ lambda ) @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ real ) ) @ B ) @ ( s @ num @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1348]) ).

thf(1308,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_SUBu_RZERO) ).

thf(10902,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_sub ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1308]) ).

thf(3103,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_ADDu_ACu_conjunct0) ).

thf(21391,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) )
      = ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( cart @ ( cart @ real @ A ) @ B ) ) ) @ matrixu_add ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ b ) ) ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ a ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3103]) ).

thf(1572,axiom,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
        <=> ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHASu_SIZEu_NUMSEGu_LT) ).

thf(12694,plain,
    ! [A: $i] :
      ( ! [B: $i,C: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
           => ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) ) )
          & ( ? [D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ num @ bool ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ bool @ ( fun @ num @ bool ) ) ) @ setspec ) @ ( s @ num @ C ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ D ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ hasu_size ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ num @ bool ) @ ( fun @ num @ bool ) ) @ gspec ) @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ A ) @ ( s @ num @ B ) ) ) ) ) ) ) @ ( s @ num @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1572]) ).

thf(343,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LEu_RMULu_EQ) ).

thf(5255,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ C ) ) ) )
     => ( ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
        = ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).

thf(1428,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
    <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNORMu_LTu_SQUARE) ).

thf(11822,plain,
    ! [A: $i,B: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
       => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ real @ a0 ) ) ) )
          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ num @ real ) ) @ realu_pow ) @ ( s @ real @ a0 ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit0 ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ vectoru_norm ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ) ) @ ( s @ real @ a0 ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1428]) ).

thf(1046,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ C ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDOTu_BASISu_BASISu_UNEQUAL) ).

thf(9303,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( s @ num @ B )
       != ( s @ num @ C ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ real ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ real ) ) @ dot ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ num @ ( cart @ real @ A ) ) @ basis ) @ ( s @ num @ C ) ) ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1046]) ).

thf(2364,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMATRIXu_VECTORu_MULu_SUBu_LDISTRIB) ).

thf(17256,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ B ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ vectoru_sub ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ D ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ B ) @ A ) @ ( fun @ ( cart @ real @ B ) @ ( cart @ real @ A ) ) ) @ matrixu_vectoru_mul ) @ ( s @ ( cart @ ( cart @ real @ B ) @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ B ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2364]) ).

thf(852,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aTREALu_MULu_LINV) ).

thf(8264,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ bool ) ) @ trealu_eq ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) ) @ trealu_mul ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ ( prod @ hreal @ hreal ) @ ( prod @ hreal @ hreal ) ) @ trealu_inv ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ A ) ) ) ) ) @ ( s @ ( prod @ hreal @ hreal ) @ ( i @ ( s @ ( fun @ num @ ( prod @ hreal @ hreal ) ) @ trealu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[852]) ).

thf(2236,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDISTu_TRIANGLEu_ALT) ).

thf(16654,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ real ) @ distance ) @ ( s @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ ( prod @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) ) @ c_ ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ D ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2236]) ).

thf(1316,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
        <=> ? [H: $i,I: $i] :
              ( ( ( p @ ( s @ bool @ I ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ A @ I ) ) ) )
                <=> ? [J: $i,K: $i] :
                      ( ( ( p @ ( s @ bool @ K ) )
                      <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                          & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ I ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ A @ J ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                    & ! [J: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                       => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
                          = ( s @ num @ I ) ) ) )
                 => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_MULTICOUNT) ).

thf(10949,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i,G: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) )
           => ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) ) )
          & ( ? [H: $i,I: $i] :
                ( ( ( p @ ( s @ bool @ I ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ H ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) @ ( s @ A @ F ) ) ) @ ( s @ B @ H ) ) ) ) )
                 => ( p @ ( s @ bool @ I ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ B @ ( fun @ bool @ ( fun @ B @ bool ) ) ) @ setspec ) @ ( s @ B @ G ) ) ) @ ( s @ bool @ I ) ) ) @ ( s @ B @ H ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ E ) ) ) @ ( s @ A @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ gspec ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ A @ G ) ) ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ A @ I ) ) ) )
                   => ? [J: $i,K: $i] :
                        ( ( ( p @ ( s @ bool @ K ) )
                         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                        & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ H ) ) ) ) )
                         => ( p @ ( s @ bool @ K ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ I ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ A @ J ) ) ) ) ) )
                  & ( ? [J: $i,K: $i] :
                        ( ( ( p @ ( s @ bool @ K ) )
                         => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ H ) ) ) ) ) )
                        & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ J ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) @ ( s @ A @ J ) ) ) @ ( s @ B @ H ) ) ) ) )
                         => ( p @ ( s @ bool @ K ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ I ) ) ) @ ( s @ bool @ K ) ) ) @ ( s @ A @ J ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ G ) ) ) @ ( s @ B @ H ) ) ) @ ( s @ A @ I ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i,I: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                    & ! [J: $i] :
                        ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ in ) @ ( s @ B @ J ) ) ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) )
                       => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ num ) @ card ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) ) @ E ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) @ ( s @ B @ J ) ) ) ) ) ) )
                          = ( s @ num @ I ) ) ) )
                 => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( fun @ A @ num ) ) ) @ D ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ bool ) ) @ F ) ) ) ) )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ I ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ num ) @ card ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1316]) ).

thf(2713,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_gt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_GT) ).

thf(19193,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_gt ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_lt ) @ ( s @ int @ B ) ) ) @ ( s @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2713]) ).

thf(1984,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i,L: $i,M: $i,N: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ u_guardedu_pattern ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ H ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ F @ bool ) ) @ geq ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i,L: $i,M: $i,N: $i] :
              ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) )
              = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i,M: $i,N: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                    & ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) )
                      = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i,M: $i,N: $i] :
                      ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) )
                      = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) )
                 => ! [K: $i] :
                      ( ! [L: $i,M: $i,N: $i,O: $i,P: $i,Q: $i] :
                          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) )
                        <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) )
                            & ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) )
                              = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) )
                            & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) ) ) )
                     => ! [L: $i] :
                          ( ! [M: $i,N: $i,O: $i,P: $i,Q: $i] :
                              ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ C @ D ) @ Q ) ) )
                              = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) )
                         => ! [M: $i,N: $i,O: $i,P: $i,Q: $i,R: $i,S: $i,T: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ P ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ R ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ Q ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ S ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ R ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ Q ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ Q ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ T ) ) ) ) )
                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ T ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_GUARDEDu_PATTERN) ).

thf(15305,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i,L: $i,M: $i,N: $i] :
          ( ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) )
          = ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ u_guardedu_pattern ) @ ( s @ bool @ ( i @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ B @ ( fun @ B @ bool ) ) @ geq ) @ ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ H ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) @ ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) @ ( s @ bool @ ( i @ ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ F @ ( fun @ F @ bool ) ) @ geq ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) @ ( s @ F @ ( i @ ( s @ ( fun @ E @ F ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i,L: $i,M: $i,N: $i] :
              ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) )
              = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i,M: $i,N: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                   => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) )
                        = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) )
                  & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) )
                      & ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) )
                        = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ C @ D ) @ M ) ) ) @ ( s @ E @ N ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i,M: $i,N: $i] :
                      ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) )
                      = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) ) ) @ ( s @ ( fun @ C @ D ) @ N ) ) ) )
                 => ! [K: $i] :
                      ( ! [L: $i,M: $i,N: $i,O: $i,P: $i,Q: $i] :
                          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) )
                           => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) ) ) )
                          & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) )
                              & ( ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) )
                                = ( s @ B @ ( i @ ( s @ ( fun @ E @ B ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) )
                              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) ) )
                           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ O ) ) ) @ ( s @ ( fun @ C @ D ) @ P ) ) ) @ ( s @ E @ Q ) ) ) ) ) )
                     => ! [L: $i] :
                          ( ! [M: $i,N: $i,O: $i,P: $i,Q: $i] :
                              ( ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ C @ D ) @ Q ) ) )
                              = ( s @ ( fun @ E @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ K ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ P ) ) ) @ ( s @ ( fun @ C @ D ) @ Q ) ) ) )
                         => ! [M: $i,N: $i,O: $i,P: $i,Q: $i,R: $i,S: $i,T: $i] :
                              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ P ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ R ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ Q ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ S ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ R ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ Q ) ) ) )
                                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ L ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ Q ) ) ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ T ) ) ) ) )
                             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ C @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ E @ A ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ C @ ( fun @ A @ bool ) ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ N ) ) ) @ ( s @ ( fun @ E @ A ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) ) ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ P ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ B ) ) @ R ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ bool ) ) @ Q ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ S ) ) ) @ ( s @ ( fun @ ( fun @ C @ D ) @ ( fun @ E @ F ) ) @ T ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1984]) ).

thf(2043,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ diff ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ diff ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUPPORTu_CLAUSESu_conjunct5) ).

thf(15629,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ diff ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) )
      = ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ diff ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) ) ) @ support ) @ ( s @ ( fun @ A @ ( fun @ A @ A ) ) @ op ) ) ) @ ( s @ ( fun @ B @ A ) @ C ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2043]) ).

thf(2213,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aHREALu_LEu_TOTAL) ).

thf(16543,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ A ) ) ) @ ( s @ hreal @ B ) ) ) )
      | ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ hreal @ bool ) @ ( i @ ( s @ ( fun @ hreal @ ( fun @ hreal @ bool ) ) @ hrealu_le ) @ ( s @ hreal @ B ) ) ) @ ( s @ hreal @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2213]) ).

thf(2320,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ ( fun @ A @ num ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ F ) ) ) @ ( s @ B @ G ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ ( fun @ A @ num ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) )
                 => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ F ) ) ) ) )
                    = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ F ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNSUMu_SWAP) ).

thf(17035,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i,F: $i] :
          ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ ( fun @ A @ num ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) ) ) @ ( s @ B @ E ) ) ) @ ( s @ A @ F ) ) )
          = ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ D ) @ ( s @ A @ F ) ) ) @ ( s @ B @ E ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i,G: $i] :
              ( ( s @ num @ ( i @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ F ) ) ) @ ( s @ B @ G ) ) )
              = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ B @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ ( fun @ A @ num ) ) ) @ C ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ F ) ) ) @ ( s @ B @ G ) ) ) ) ) )
         => ! [E: $i] :
              ( ! [F: $i,G: $i,H: $i] :
                  ( ( s @ num @ ( i @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) ) ) @ ( s @ A @ H ) ) )
                  = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ G ) @ ( s @ A @ H ) ) ) ) ) )
             => ! [F: $i,G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) ) )
                 => ( ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ A @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ A @ num ) ) ) @ E ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ F ) ) ) ) )
                    = ( s @ num @ ( i @ ( s @ ( fun @ ( fun @ B @ num ) @ num ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ num ) @ num ) ) @ nsum ) @ ( s @ ( fun @ B @ bool ) @ H ) ) ) @ ( s @ ( fun @ B @ num ) @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ ( fun @ B @ num ) ) @ ( fun @ B @ num ) ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ ( fun @ B @ num ) ) @ F ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2320]) ).

thf(1664,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                  & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aDIFF) ).

thf(13306,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i,D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) )
                    & ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ F ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ bool @ ( fun @ A @ bool ) ) ) @ setspec ) @ ( s @ A @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ A @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ A @ E ) ) ) ) ) )
     => ! [C: $i,D: $i] :
          ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ diff ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) )
          = ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ gspec ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ B ) @ ( s @ ( fun @ A @ bool ) @ C ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1664]) ).

thf(1843,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aORTHOGONALu_SYM) ).

thf(14450,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) @ ( s @ ( cart @ real @ A ) @ C ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ orthogonal ) @ ( s @ ( cart @ real @ A ) @ C ) ) ) @ ( s @ ( cart @ real @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1843]) ).

thf(2256,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_SYM) ).

thf(16760,plain,
    ! [A: $i,B: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2256]) ).

thf(486,axiom,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_POWu_LE) ).

thf(6141,plain,
    ! [A: $i,B: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ A ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ num @ int ) ) @ intu_pow ) @ ( s @ int @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).

thf(2596,axiom,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_ABS) ).

thf(18620,plain,
    ! [A: $i,B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ int @ bool ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ bool ) ) @ intu_le ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ B ) ) ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_abs ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2596]) ).

thf(2244,axiom,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ rows ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aFINITEu_ROWS) ).

thf(16681,plain,
    ! [A: $i,B: $i,C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( cart @ real @ A ) @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ ( cart @ real @ A ) @ bool ) @ ( i @ ( s @ ( fun @ ( cart @ ( cart @ real @ A ) @ B ) @ ( fun @ ( cart @ real @ A ) @ bool ) ) @ rows ) @ ( s @ ( cart @ ( cart @ real @ A ) @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2244]) ).

thf(1201,axiom,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aVECTORu_NEGu_MINUS1) ).

thf(10192,plain,
    ! [A: $i] :
      ( ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ vectoru_neg ) @ ( s @ ( cart @ real @ A ) @ x ) ) )
      = ( s @ ( cart @ real @ A ) @ ( i @ ( s @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( cart @ real @ A ) @ ( cart @ real @ A ) ) ) @ r_ ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_neg ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) ) ) ) @ ( s @ ( cart @ real @ A ) @ x ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1201]) ).

thf(1007,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
        <=> ? [F: $i,G: $i] :
              ( ( ( p @ ( s @ bool @ G ) )
              <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q0 ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
              & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
            <=> ? [G: $i,H: $i] :
                  ( ( ( p @ ( s @ bool @ H ) )
                  <=> ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                      & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                & ! [G: $i,H: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aCARDu_LEu_SUBPOWERSET) ).

thf(9087,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i,E: $i] :
          ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) )
           => ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q0 ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q0 ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
          & ( ? [F: $i,G: $i] :
                ( ( ( p @ ( s @ bool @ G ) )
                 => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q0 ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) )
                & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q0 ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
                 => ( p @ ( s @ bool @ G ) ) )
                & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ B @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) @ ( s @ bool @ G ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ D ) ) ) @ ( s @ ( fun @ B @ bool ) @ E ) ) ) ) ) )
     => ! [D: $i] :
          ( ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) )
               => ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
              & ( ? [G: $i,H: $i] :
                    ( ( ( p @ ( s @ bool @ H ) )
                     => ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) ) )
                    & ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ subset ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
                     => ( p @ ( s @ bool @ H ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ bool @ ( fun @ ( fun @ A @ bool ) @ bool ) ) ) @ setspec ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) @ ( s @ bool @ H ) ) ) @ ( s @ ( fun @ A @ bool ) @ G ) ) ) ) )
               => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ A @ bool ) @ F ) ) ) ) ) )
         => ! [E: $i,F: $i] :
              ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) )
                & ! [G: $i,H: $i] :
                    ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ p0 ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) )
                   => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ q0 ) @ ( s @ ( fun @ B @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ image ) @ ( s @ ( fun @ A @ B ) @ G ) ) ) @ ( s @ ( fun @ A @ bool ) @ H ) ) ) ) ) ) ) )
             => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ bool ) ) @ l_a_u_c ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ D ) @ ( s @ ( fun @ A @ bool ) @ E ) ) ) ) ) ) ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ gspec ) @ ( s @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ C ) @ ( s @ ( fun @ B @ bool ) @ F ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1007]) ).

thf(1892,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i] :
          ( ( s @ F @ ( i @ ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) )
          = ( s @ F @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ F ) @ ( i @ ( s @ ( fun @ E @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ F ) ) @ u_match ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i] :
              ( ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) )
              = ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i,M: $i] :
                  ( ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ K ) ) ) @ ( s @ ( fun @ B @ C ) @ L ) ) ) @ ( s @ D @ M ) ) )
                  = ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) @ ( s @ ( fun @ B @ C ) @ L ) ) ) @ ( s @ D @ M ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ K ) @ ( s @ ( fun @ B @ C ) @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i,M: $i] :
                      ( ( s @ ( fun @ D @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ ( fun @ B @ C ) @ M ) ) )
                      = ( s @ ( fun @ D @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) )
                 => ! [K: $i,L: $i,M: $i,N: $i,O: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ D @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ N ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ D @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ N ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ D @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ O ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aADMISSIBLEu_MATCH) ).

thf(14691,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ! [H: $i,I: $i,J: $i,K: $i] :
          ( ( s @ F @ ( i @ ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) ) ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) )
          = ( s @ F @ ( i @ ( s @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ F ) @ ( i @ ( s @ ( fun @ E @ ( fun @ ( fun @ E @ ( fun @ F @ bool ) ) @ F ) ) @ u_match ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ H ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ I ) @ ( s @ ( fun @ B @ C ) @ J ) ) ) @ ( s @ D @ K ) ) ) ) ) )
     => ! [H: $i] :
          ( ! [I: $i,J: $i,K: $i] :
              ( ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) )
              = ( s @ ( fun @ D @ F ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ G ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ I ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ B @ C ) @ K ) ) ) )
         => ! [I: $i] :
              ( ! [J: $i,K: $i,L: $i,M: $i] :
                  ( ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ D @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ K ) ) ) @ ( s @ ( fun @ B @ C ) @ L ) ) ) @ ( s @ D @ M ) ) )
                  = ( s @ ( fun @ F @ bool ) @ ( i @ ( s @ ( fun @ E @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ J ) @ ( s @ ( fun @ B @ C ) @ L ) ) ) @ ( s @ D @ M ) ) ) @ ( s @ E @ ( i @ ( s @ ( fun @ D @ E ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ K ) @ ( s @ ( fun @ B @ C ) @ L ) ) ) @ ( s @ D @ M ) ) ) ) ) )
             => ! [J: $i] :
                  ( ! [K: $i,L: $i,M: $i] :
                      ( ( s @ ( fun @ D @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ ( fun @ B @ C ) @ M ) ) )
                      = ( s @ ( fun @ D @ ( fun @ F @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) ) @ I ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ L ) ) ) @ ( s @ ( fun @ B @ C ) @ M ) ) ) )
                 => ! [K: $i,L: $i,M: $i,N: $i,O: $i] :
                      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ D @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ N ) ) ) )
                        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ D @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ F @ bool ) ) ) ) ) @ J ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ O ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ N ) ) ) ) ) ) )
                     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ ( fun @ ( fun @ D @ A ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ bool ) ) ) ) @ admissible ) @ ( s @ ( fun @ B @ ( fun @ A @ bool ) ) @ K ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ bool ) ) @ L ) ) ) @ ( s @ ( fun @ D @ A ) @ M ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) @ ( i @ ( s @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ ( fun @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ F ) ) ) ) @ H ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ E ) ) @ N ) ) ) @ ( s @ ( fun @ ( fun @ B @ C ) @ ( fun @ D @ ( fun @ E @ ( fun @ F @ bool ) ) ) ) @ O ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1892]) ).

thf(1434,axiom,
    ! [A: $i] :
      ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
    <=> ( ( s @ int @ A )
        = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SGNu_EQu_conjunct0) ).

thf(11858,plain,
    ! [A: $i] :
      ( ( ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) )
      & ( ( ( s @ int @ A )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) )
       => ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ intu_sgn ) @ ( s @ int @ A ) ) )
          = ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1434]) ).

thf(789,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_ABSu_INFu_LE) ).

thf(7893,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( s @ ( fun @ real @ bool ) @ A )
         != ( s @ ( fun @ real @ bool ) @ empty ) )
        & ! [C: $i] :
            ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ ( fun @ real @ bool ) @ bool ) ) @ in ) @ ( s @ real @ C ) ) ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) )
           => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ realu_abs ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ real @ bool ) @ real ) @ inf ) @ ( s @ ( fun @ real @ bool ) @ A ) ) ) ) ) ) ) @ ( s @ real @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[789]) ).

thf(947,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aRIGHTu_ADDu_DISTRIB) ).

thf(8818,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) ) @ ( s @ num @ C ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[947]) ).

thf(2865,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTu_ADD2) ).

thf(20039,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2865]) ).

thf(1691,axiom,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i] :
        ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aNADDu_MULu_LINVu_LEMMA8) ).

thf(13478,plain,
    ! [A: $i] :
      ( ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ nadd @ bool ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ nadd @ bool ) ) @ naddu_eq ) @ ( s @ nadd @ A ) ) ) @ ( s @ nadd @ ( i @ ( s @ ( fun @ num @ nadd ) @ naddu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) )
     => ? [B: $i] :
        ! [C: $i,D: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ num ) @ dist ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ D ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ nadd @ ( fun @ num @ num ) ) @ naddu_rinv ) @ ( s @ nadd @ A ) ) ) @ ( s @ num @ C ) ) ) ) ) ) ) ) ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ p_ ) @ ( s @ num @ C ) ) ) @ ( s @ num @ D ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1691]) ).

thf(2259,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aLEu_TRANS) ).

thf(16775,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ num @ bool ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ bool ) ) @ l_a_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2259]) ).

thf(2635,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LETu_TRANS) ).

thf(18798,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ B ) ) ) @ ( s @ real @ C ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2635]) ).

thf(2423,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ univ ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aUNIONu_UNIVu_conjunct0) ).

thf(17636,plain,
    ! [A: $i,B: $i] :
      ( ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ union ) @ ( s @ ( fun @ A @ bool ) @ univ ) ) ) @ ( s @ ( fun @ A @ bool ) @ B ) ) )
      = ( s @ ( fun @ A @ bool ) @ univ ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2423]) ).

thf(71,axiom,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_MULu_RID) ).

thf(3575,plain,
    ! [A: $i] :
      ( ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_mul ) @ ( s @ real @ A ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ num @ real ) @ realu_ofu_num ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ bit1 ) @ ( s @ num @ u_0 ) ) ) ) ) ) ) ) )
      = ( s @ real @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(889,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aSUMu_CLAUSESu_conjunct1) ).

thf(8465,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ finite ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) )
     => ( ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ insert ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) )
        = ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ real @ ( fun @ real @ real ) ) ) @ cond ) @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ in ) @ ( s @ A @ B ) ) ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ ( i @ ( s @ ( fun @ A @ real ) @ C ) @ ( s @ A @ B ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ ( fun @ A @ real ) @ real ) @ ( i @ ( s @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ real ) @ real ) ) @ sum ) @ ( s @ ( fun @ A @ bool ) @ D ) ) ) @ ( s @ ( fun @ A @ real ) @ C ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[889]) ).

thf(1788,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aREALu_LTEu_ADD2) ).

thf(14129,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ A ) ) ) @ ( s @ real @ B ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_le ) @ ( s @ real @ C ) ) ) @ ( s @ real @ D ) ) ) ) )
     => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ real @ bool ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ bool ) ) @ realu_lt ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ A ) ) ) @ ( s @ real @ C ) ) ) ) ) @ ( s @ real @ ( i @ ( s @ ( fun @ real @ real ) @ ( i @ ( s @ ( fun @ real @ ( fun @ real @ real ) ) @ realu_add ) @ ( s @ real @ B ) ) ) @ ( s @ real @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1788]) ).

thf(2824,axiom,
    ! [A: $i] :
      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
    <=> ! [B: $i] :
          ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
        <=> ? [C: $i] :
              ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
              & ! [D: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
                 => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aWFu_EQ) ).

thf(19794,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) )
       => ! [B: $i] :
            ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
             => ? [C: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
                  & ! [D: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
                     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) )
            & ( ? [C: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
                  & ! [D: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
                     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) )
             => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) ) )
      & ( ! [B: $i] :
            ( ( ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
             => ? [C: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
                  & ! [D: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
                     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) ) )
            & ( ? [C: $i] :
                  ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) )
                  & ! [D: $i] :
                      ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ ( i @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) @ ( s @ A @ D ) ) ) @ ( s @ A @ C ) ) ) )
                     => ~ ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ D ) ) ) ) ) )
             => ? [C: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ A @ bool ) @ B ) @ ( s @ A @ C ) ) ) ) ) )
       => ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ bool ) @ wf ) @ ( s @ ( fun @ A @ ( fun @ A @ bool ) ) @ l_l_ ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2824]) ).

thf(198,axiom,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMULTu_CLAUSESu_conjunct0) ).

thf(4298,plain,
    ! [A: $i] :
      ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) ) @ ( s @ num @ A ) ) )
      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(1088,axiom,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ bool ) @ numu_coprime ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ bool ) @ intu_coprime ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',anumu_coprime) ).

thf(9519,plain,
    ! [A: $i,B: $i] :
      ( ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ num @ num ) @ bool ) @ numu_coprime ) @ ( s @ ( prod @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( prod @ num @ num ) ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ ( prod @ num @ num ) ) ) @ c_ ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) ) ) )
      = ( s @ bool @ ( i @ ( s @ ( fun @ ( prod @ int @ int ) @ bool ) @ intu_coprime ) @ ( s @ ( prod @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( prod @ int @ int ) ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ ( prod @ int @ int ) ) ) @ c_ ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ A ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ num @ int ) @ intu_ofu_num ) @ ( s @ num @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1088]) ).

thf(90,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aINTu_SUBu_LDISTRIB) ).

thf(3669,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ B ) ) ) @ ( s @ int @ C ) ) ) ) )
      = ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_sub ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ B ) ) ) ) ) @ ( s @ int @ ( i @ ( s @ ( fun @ int @ int ) @ ( i @ ( s @ ( fun @ int @ ( fun @ int @ int ) ) @ intu_mul ) @ ( s @ int @ A ) ) ) @ ( s @ int @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(1636,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( s @ num @ A )
          = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
    <=> ? [C: $i] :
          ( ( ( p @ ( s @ bool @ C ) )
          <=> ( ( s @ num @ A )
              = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
          & ? [D: $i] :
              ( ( ( p @ ( s @ bool @ D ) )
              <=> ( ( s @ num @ B )
                  = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
              & ? [E: $i] :
                  ( ( ( p @ ( s @ bool @ E ) )
                  <=> ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
                      = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                  & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ C ) ) ) @ ( s @ bool @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aMODu_EXISTS) ).

thf(13107,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) )
       => ? [C: $i] :
            ( ( ( p @ ( s @ bool @ C ) )
             => ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
            & ( ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
             => ( p @ ( s @ bool @ C ) ) )
            & ? [D: $i] :
                ( ( ( p @ ( s @ bool @ D ) )
                 => ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                & ( ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( p @ ( s @ bool @ D ) ) )
                & ? [E: $i] :
                    ( ( ( p @ ( s @ bool @ E ) )
                     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
                        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                    & ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
                        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                     => ( p @ ( s @ bool @ E ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ C ) ) ) @ ( s @ bool @ E ) ) ) ) ) ) ) )
      & ( ? [C: $i] :
            ( ( ( p @ ( s @ bool @ C ) )
             => ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
            & ( ( ( s @ num @ A )
                = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
             => ( p @ ( s @ bool @ C ) ) )
            & ? [D: $i] :
                ( ( ( p @ ( s @ bool @ D ) )
                 => ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                & ( ( ( s @ num @ B )
                    = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                 => ( p @ ( s @ bool @ D ) ) )
                & ? [E: $i] :
                    ( ( ( p @ ( s @ bool @ E ) )
                     => ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
                        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) ) )
                    & ( ( ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ mod ) @ ( s @ num @ A ) ) ) @ ( s @ num @ B ) ) )
                        = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ numeral ) @ ( s @ num @ u_0 ) ) ) )
                     => ( p @ ( s @ bool @ E ) ) )
                    & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( i @ ( s @ ( fun @ bool @ ( fun @ bool @ ( fun @ bool @ bool ) ) ) @ cond ) @ ( s @ bool @ D ) ) ) @ ( s @ bool @ C ) ) ) @ ( s @ bool @ E ) ) ) ) ) ) )
       => ? [C: $i] :
            ( ( s @ num @ A )
            = ( s @ num @ ( i @ ( s @ ( fun @ num @ num ) @ ( i @ ( s @ ( fun @ num @ ( fun @ num @ num ) ) @ t_ ) @ ( s @ num @ B ) ) ) @ ( s @ num @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1636]) ).

thf(2486,axiom,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ f0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ t ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',aboolu_INDUCT) ).

thf(17982,plain,
    ! [A: $i] :
      ( ( ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ f0 ) ) ) )
        & ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ t ) ) ) ) )
     => ! [B: $i] : ( p @ ( s @ bool @ ( i @ ( s @ ( fun @ bool @ bool ) @ A ) @ ( s @ bool @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2486]) ).

thf(21744,plain,
    $false,
    inference(cvc4,[status(thm)],[14404,19117,13950,16466,18180,5385,5810,19204,15767,21624,7304,10007,21729,20256,18768,8960,15170,9405,17507,13024,14885,17639,5469,10647,19672,20690,17803,5116,13162,7272,13199,21175,5088,12597,12710,12490,7890,15278,4005,10785,5136,8154,15720,21710,7569,15418,4262,6976,20014,3399,18504,20850,6544,4169,6712,13643,21439,3666,9180,15472,19594,18442,20902,19623,3979,8507,20464,16808,20559,11581,21450,14220,3863,12440,3830,20683,14139,17175,17998,8266,3449,10081,9864,6986,8646,18228,10363,6042,16124,5733,11318,8475,19745,18154,19611,20726,12507,18311,14944,8077,11659,17920,15380,19187,11790,16950,9032,18985,18995,12371,19533,19213,15532,19687,3534,9342,14296,15019,8855,12634,6063,15980,6576,10299,9573,13834,18020,17310,11256,14976,12129,19088,12649,6918,8434,4339,13007,13712,21114,14625,6495,7008,20922,3905,8797,20813,15267,7641,12084,17278,16619,5000,20136,21418,7132,20268,16566,3285,17641,20400,9324,18549,3460,13159,6900,8463,19438,3912,21111,10705,13462,16280,15827,14729,21072,9337,20534,8447,8169,5224,20626,17847,21196,12911,9191,6426,12261,18635,11266,12896,4326,17815,18546,16845,3402,17715,18837,10463,11881,19964,15579,16229,4487,9268,13202,20477,21538,20305,5589,7484,21016,18861,17411,18178,6971,13660,14617,19230,10775,15357,13187,18592,5037,3794,8182,10484,14347,14264,19291,9526,16982,10433,5565,16777,12852,19900,16513,5906,13675,14124,10772,10395,14519,13457,6665,10358,20293,17594,12479,15632,8879,18824,14273,13276,14694,21229,14665,14726,18897,20352,8667,15217,16283,18921,15948,6655,19787,7345,8298,21627,3616,6515,6851,3781,15175,9218,12725,15034,9000,6078,15885,14149,9356,21403,16376,13960,6551,3481,10726,3562,6996,7068,15051,13298,18729,7452,17577,8838,18133,17375,6804,17126,16965,8204,20986,15207,21096,11756,18800,14591,7748,3895,14354,20885,16164,18595,20011,15735,8303,10351,3766,8133,11250,15683,11171,12447,7168,20034,8851,21467,16933,4792,13452,16491,8848,13628,20556,11436,17492,9064,16733,12916,3915,6268,16456,10022,16304,20569,10616,11542,4496,15202,10908,10474,8539,14483,16623,20166,18834,15723,11699,18040,9027,3777,9379,12757,13392,9159,10078,18992,9994,3308,12646,4124,7326,14559,5056,5770,18420,8866,8671,18938,21415,13571,20845,5301,21560,9971,18252,4030,18120,19294,8330,16979,13361,14644,11912,8504,7761,21239,11831,13923,9013,16351,17888,6182,20441,18717,16118,17197,10633,21124,7993,13039,17925,6707,7701,21530,12843,12254,9960,7496,3820,18437,21143,3204,20707,3442,18754,5765,9300,19267,4546,16332,5429,16626,9989,20191,18760,3195,17117,9287,11976,20541,3644,4896,17610,6039,10587,13803,9410,6538,8426,10089,13091,20340,19342,6547,10751,9216,9561,9084,7683,8357,19108,14489,17751,11010,6423,6066,11966,5274,10142,10221,7945,15881,7877,9360,8522,11111,18782,20704,3542,4275,4349,8069,9315,11812,10010,18030,6113,10337,5640,12606,9805,5989,6332,13273,19525,19710,4428,7743,6208,12869,5836,14228,20508,20418,21268,7011,20232,21220,11981,3216,16740,14487,7754,21394,21367,19052,16391,15819,19780,5338,15601,18380,19589,16757,17097,11331,15214,21491,12291,9480,15028,19658,15586,6905,14455,15477,20364,16772,18292,11321,21447,7209,15742,11564,4179,16035,14103,13580,8468,18659,16873,21550,4622,11378,6272,20203,15509,19120,17962,9529,4675,10650,19937,18982,7047,5942,16545,17181,8674,6098,20181,15744,10570,4265,9039,17603,6828,4383,20259,15167,3557,5493,15193,6734,13350,8252,17004,3866,20675,7032,8558,20790,11495,11139,3268,13715,7500,6382,9266,17591,13788,7733,7236,18370,7226,21249,10497,17302,6983,17979,8934,12236,17652,18773,9636,20579,15105,6081,13356,15878,9854,4988,9204,8394,5264,10729,21582,18407,20890,9593,16151,12018,19643,6840,9283,11778,4329,20499,11064,15274,4903,16929,14697,19338,4018,20285,18075,12051,21723,7247,13538,8860,18111,10271,14132,8802,6523,14532,14423,20997,19201,16463,8791,8929,19598,17916,14547,21510,3525,3438,6187,21352,11215,17090,6176,21322,12105,16840,3621,3973,8659,17398,10491,7677,5914,18126,14309,21427,21656,15522,11709,11398,3569,14066,9680,11793,4798,11446,15975,8246,13543,6157,11971,12582,21303,9781,12041,15834,18786,11036,18037,16897,16785,19617,16204,6591,10538,4707,18475,7413,19717,8050,7707,4718,6809,16007,21069,11312,5773,18123,11325,14987,20934,10145,4033,7931,9796,8261,8774,20262,20619,9369,19608,5643,6110,5375,19177,3685,8306,5648,8406,7336,14113,14874,8141,11136,11806,4364,15008,4996,10405,19998,18392,3676,5846,13072,5343,8686,4924,9035,13521,10057,5897,19947,16976,20008,16136,17571,9187,21520,4536,6021,19085,13600,10801,14798,9236,3327,16577,9347,4701,18656,3226,14637,11505,6327,19033,8443,5629,4519,18814,11853,21232,13216,20905,11195,7722,9896,4483,15958,12762,5695,13683,18412,12898,8655,8397,3418,14569,10494,7197,21009,9556,20762,3229,20871,14445,21202,10943,18909,19401,9682,5875,10811,20602,13386,11993,4893,3591,6482,8927,20528,11395,4888,8233,19334,13019,20216,7725,6731,9968,20253,4480,18809,3265,12457,17389,20121,14420,20480,10139,13709,21399,21223,19874,17655,12029,18498,15751,15642,7859,15364,19903,7885,11984,13104,7937,5562,7357,17242,17289,7269,7410,10911,21444,10258,9164,5487,15258,19319,8842,12132,10639,17363,16780,18796,20370,4031,10477,9041,15542,4172,4999,9881,10345,3679,17178,4434,20057,13094,20485,15082,7730,17574,19595,21507,3735,20848,18493,4712,16399,16973,9296,8778,21310,9522,16729,19987,14501,9861,14357,19305,8994,11768,19099,5002,12510,6264,20931,9817,5512,20989,12737,7229,18864,11719,20807,19967,11587,16215,11223,19122,5956,18784,11865,5815,14222,20152,11217,16484,7710,14973,21579,10657,9207,17683,15531,18151,20188,8550,7084,14524,9981,9126,12846,12687,6915,4499,12715,4493,5085,20550,3816,7984,13498,5893,17876,3814,14648,11212,3207,7244,16307,11074,3682,4021,9734,11371,13583,20521,3889,17945,21083,15298,15595,9177,3282,16652,15670,8810,18415,10710,17738,8255,17200,13654,19751,5702,11570,19390,10939,19980,5655,14541,16371,13282,8026,8700,7185,9058,14248,6848,12279,3198,19748,18632,16790,15383,13612,15994,4994,14430,18283,9586,17606,3578,15665,15093,8736,13568,20721,7503,6292,14681,12881,7625,19322,11629,16497,19190,9391,20197,12618,7545,20876,21732,5113,6096,4979,17891,21066,12849,16793,7934,12603,16252,11948,5658,7853,8226,13681,21187,8031,9885,14100,11702,13515,3647,18108,21576,7767,3931,3408,4680,8403,5330,9641,20244,3799,7686,4027,4063,16899,6709,12760,12264,13872,8429,18971,15559,7923,18204,5626,3699,8530,7799,11356,6339,11734,12987,6324,15739,5053,5221,10748,8545,17306,14327,12404,16588,18961,12538,15714,19864,13347,6940,12792,5270,7249,15078,6335,15847,20787,10986,9073,15454,21325,5345,6691,16968,10791,11884,10281,6955,9245,21234,6603,9945,21533,20678,16629,6459,16108,7481,17010,10566,4789,20387,11007,4636,8063,10644,15065,7838,13825,8014,7610,21275,18017,9576,20553,5583,3963,20448,18404,9525,21150,7698,3746,20570,6360,11929,11400,19341,3650,13074,21421,8662,15729,14276,16055,3440,10760,15281,8138,5475,5951,12974,14110,18672,9530,4291,19620,20175,8176,14623,3954,13928,17328,7898,3952,20979,14142,4301,15798,16847,10946,15545,19425,6967,5818,15882,18653,9910,8555,3179,12967,14566,13291,14809,19921,7717,5651,14473,3210,17001,15141,7846,7981,4167,10528,14212,6363,16201,13190,19758,16920,10342,8008,9419,16958,19373,16568,8258,4421,17074,15619,15022,5253,7604,17218,13066,4513,16656,18055,8053,12803,5213,10195,3428,8968,15270,6837,17165,11026,17780,21306,16683,7404,19358,16451,7313,18650,14131,17239,19867,12357,9429,10094,8952,4736,9645,12600,12314,18137,15852,4530,6724,19906,14561,21006,10560,14369,21543,10158,9461,4804,10720,3242,8458,12955,11828,11269,17360,12632,10423,19679,17720,8072,16468,15788,19393,3743,17873,14392,7221,5900,8410,21162,15200,18714,6475,13483,5973,5498,14509,7738,18608,20229,3390,18347,3222,10148,10207,3937,18027,19684,21119,21146,15285,20681,11315,12838,15399,10047,8191,14526,10533,16058,3628,6908,8455,16237,18934,3967,10302,12533,18802,20698,18845,8090,5801,17267,17823,12800,10180,14116,18581,7826,4069,13340,4416,13490,11927,7333,15934,18998,21270,6988,9942,14352,11000,11873,3201,16854,4361,12138,3454,3802,5872,14848,13932,7193,11031,19149,13398,8971,18640,3903,18364,15290,20303,4352,6533,15991,12751,4644,15042,17293,14821,18308,13213,15414,12282,8739,4288,17499,18159,7407,17482,8512,15499,15592,20589,3183,12889,17812,20241,21485,21441,9044,17863,16220,19198,10656,16666,9008,15535,19227,15624,7397,19630,20096,11987,8279,16528,17984,17767,16644,16048,15136,16669,16329,13196,8955,14092,18757,9171,18357,19090,5209,5930,16886,11328,13879,16990,14362,12913,20004,5887,13480,20793,21214,18389,21434,15344,11539,18169,19315,16923,7802,10190,20614,19775,9226,17646,9256,19283,12677,19977,18924,21023,3807,18472,10592,4919,5808,17321,6469,16343,4921,11578,12609,20226,16265,21349,8333,14231,21165,15062,11692,16477,17050,5524,6507,18255,9416,12874,10175,9765,4380,16709,15057,19530,16857,19846,15570,9502,15707,18968,20490,7155,7831,16260,16505,14620,12119,20466,13259,14982,18709,8480,12683,16747,3262,10993,19418,6001,12704,15025,17563,5016,10274,20701,13336,17991,5979,13849,17949,20384,9650,3696,6492,16012,18242,20024,17703,7372,5043,13296,17748,15696,13559,8807,20657,8495,10425,18064,6870,6510,3519,16154,15079,19814,18689,16750,16935,10155,4662,13338,3636,14097,7736,9602,17280,18302,12317,7487,5121,11809,11936,5923,18990,7223,9025,7751,7440,17141,20847,13395,3758,3945,16323,4677,17573,12541,18418,4801,20411,17315,16209,4709,12840,7041,19381,4605,5050,19083,13561,7704,12259,19296,9057,10557,3604,16649,20475,20237,14642,9339,4612,16349,20586,19264,21497,18213,9820,12011,10525,5799,12964,7622,17009,3213,18812,16291,17115,10044,18590,19881,13358,17905,8767,11819,19269,9285,14258,14076,17041,5033,4127,14144,3751,6385,13953,16796,18314,15765,16334,13217,6937,6357,17368,5011,16515,15649,8286,10083,17162,19234,8862,7719,11951,11502,17077,8665,3663,4672,17789,15517,6991,16945,12524,20581,10348,4413,10051,11662,14134,18822,8492,8300,11968,16823,12748,19466,20609,18444,17918,15485,7455,15887,3619,5539,11973,3641,15609,15329,18003,7843,9290,6248,20864,6549,6334,16782,20518,20041,17786,14942,17287,11765,3302,7021,175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).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12  % Problem  : GEO452+1 : TPTP v8.1.2. Released v7.0.0.
% 0.07/0.15  % Command  : run_Leo-III %s %d
% 0.15/0.36  % Computer : n026.cluster.edu
% 0.15/0.36  % Model    : x86_64 x86_64
% 0.15/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36  % Memory   : 8042.1875MB
% 0.15/0.36  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36  % CPULimit : 300
% 0.15/0.36  % WCLimit  : 300
% 0.15/0.36  % DateTime : Fri May 19 04:14:06 EDT 2023
% 0.15/0.36  % CPUTime  : 
% 0.98/0.87  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 3.34/1.63  % [INFO] 	 Parsing done (764ms). 
% 3.34/1.64  % [INFO] 	 Running in sequential loop mode. 
% 3.81/1.84  % [INFO] 	 eprover registered as external prover. 
% 3.81/1.84  % [INFO] 	 cvc4 registered as external prover. 
% 3.81/1.85  % [INFO] 	 Scanning for conjecture ... 
% 6.64/2.75  % [INFO] 	 Found a conjecture and 3176 axioms. Running axiom selection ... 
% 10.21/4.59  % [INFO] 	 Axiom selection finished. Selected 3176 axioms (removed 0 axioms). 
% 19.51/11.67  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 19.80/11.79  % [INFO] 	 Type checking passed. 
% 19.80/11.79  % [CONFIG] 	 Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>.  Searching for refutation ... 
% 198.40/131.63  % External prover 'cvc4' found a proof!
% 198.40/131.63  % [INFO] 	 Killing All external provers ... 
% 198.40/131.63  % Time passed: 131107ms (effective reasoning time: 129981ms)
% 198.40/131.63  % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 198.40/131.64  % Axioms used in derivation (3176): aDIVISIONu_0, aLEu_conjunct1, aVECTORu_ADDu_SUB, aHREALu_LEu_TOTAL, aREALu_POLYu_CLAUSESu_conjunct9, aintu_mul, aPOWu_2u_SQRTu_ABS, aMATRIXu_VECTORu_MULu_LID, aINTu_MINu_ACIu_conjunct1, aEQu_EXT, aNSUMu_IMAGE, aKL, aNSUMu_ADDu_GEN, aREALu_MULu_ACu_conjunct0, aREALu_ABSu_NORM, aLEu_LT, aNSUMu_DELETE, aTRIVu_FORALLu_IMPu_THM, aREALu_MINu_ACIu_conjunct3, aCARDu_EQu_REFL, aINTu_MAXu_ACIu_conjunct4, aDISJu_ACIu_conjunct2, aINTu_MULu_LNEG, aVSUMu_NORMu_BOUND, aARITHu_SUBu_conjunct8, aODDu_ADD, aNOTu_INSERTu_EMPTY, aRIGHTu_ANDu_FORALLu_THM, aVECTORu_SUBu_COMPONENT, aMODu_NSUMu_MOD, aSUMu_SUBSET, aPOWu_2u_SQRT, aINTu_LEu_SQUAREu_ABS, aINTu_LEu_REFL, aFINITEu_BOOL, avectoru_sub, aNSUMu_TRIVu_NUMSEG, avectoru_mul, aREALu_OFu_NUMu_MIN, aUNIONSu_MONOu_IMAGE, aINTu_SGNu_CASES, aPAIREDu_ETAu_THMu_conjunct1, aoptionu_elim, aARITHu_GTu_conjunct4, aoneu_INDUCT, aBIT1, arealu_ofu_num, aLTu_MULT2, aINu_FROM, aMEMu_LISTu_OFu_SET, aINTu_ABSu_POS, aSUBSETu_UNIONu_conjunct1, aREALu_POWu_EQ, aREALu_LETu_ANTISYM, aITERATEu_CASES, aMATRIXu_EQUALu_ROWS, aCONJu_ACIu_conjunct3, aSUMu_SUMu_RESTRICT, aDISTu_RZERO, aLEMMA, aSUPERADMISSIBLEu_MATCHu_SEQPATTERN, aCollectu_geomo_PERu_MUL3u_conjunct0, aDEu_MORGANu_THM, aCARDu_LETu_TRANS, aREALu_CONVEXu_BOUNDu_LE, aDELETEu_COMM, aROWu_TRANSP, aINTu_DIFFSQ, aTREALu_OFu_NUMu_WELLDEF, aVSUMu_BIJECTION, aFILTERu_conjunct0, aCARDu_MULu_FINITE, aMATRIXu_MULu_LMUL, aREALu_LEu_INVu_EQ, aUNIONSu_UNION, asuperadmissible, aTREALu_MULu_SYM, aNADDu_LEu_ADD, aDIVu_MONO2, aintu_gcd, aSETu_OFu_LISTu_MAP, aINTERS, aINTu_NEGu_ADD, aDIMINDEXu_HASu_SIZEu_FINITEu_SUM, aORTHOGONALu_LVSUM, aINTu_SUBu_REFL, aTRIVu_EXISTSu_IMPu_THM, aFINITEu_SING, aone, aEXTENDu_INSEG, aLINEARu_VMULu_COMPONENT, aREVERSEu_REVERSE, aFINITEu_REALu_INTERVALu_conjunct1, aTREALu_INVu_WELLDEF, aORTHOGONALu_RVSUM, aFINITEu_DELETE, aNADDu_DISTu_LEMMA, aFINITEu_INDEXu_NUMBERS, aREALu_POWu_LT2u_REV, aINTu_FORALLu_ABS, aREALu_ABSu_SUBu_NORM, aCARDu_LEu_LIST, aREALu_POWu_EQu_0, aDOTu_LZERO, aITLIST2u_DEFu_conjunct0, aARITHu_SUBu_conjunct5, aITERATEu_OPu_GEN, aTREALu_ADDu_SYMu_EQ, aEXISTSu_REFL, aADMISSIBLEu_NEST, aLTu_EXP, aDISTu_POSu_LT, aADDu_0, apairwise, aCASEWISEu_conjunct0, aWOSETu_ANTISYM, aHREALu_MULu_LZERO, aNSUMu_CASES, aARITHu_ADDu_conjunct4, aBUTLASTu_conjunct0, aMATRIXu_NEGu_0, aZLu_SUBSETSu_UNIONSu_NONEMPTY, aFORALLu_COUNTABLEu_ASu_IMAGE, aSUMu_UNIONu_NONZERO, aREALu_LTu_SUPu_FINITE, aUNIONu_ACIu_conjunct1, aVECTORu_ADDu_RID, aINTu_POWu_LE, aREALu_ABSu_ZERO, aINTu_LTu_LADD, aREALu_POWu_LT2, aALLu_MEM, aSUPERADMISSIBLEu_T, aNADDu_ARCHu_ZERO, aTREALu_LEu_TRANS, aREALu_SUBu_POW, aEXPu_2, aDIMINDEXu_2, aEQu_ADDu_RCANCELu_0, aVSUMu_COMPONENT, aREALu_INVu_LEu_1, aREALu_LEu_LMUL, aINTu_POWu_MONO, aARITHu_ADDu_conjunct5, aVECTORu_ADDu_ASSOC, aINu_SUPPORT, aHREALu_ADDu_LID, aREALu_LTu_ADD1, aTREALu_LEu_WELLDEF, aREALu_LTu_LADD, aINTu_SOSu_EQu_0, aREALu_OFu_NUMu_ADD, aINTu_ABSu_SUB, aFINITEu_REALu_INTERVALu_conjunct7, aMONOu_ALL2, aFINITEu_CROSS, aTOPOLOGICALu_SORT, aREALu_INVu_LTu_1, aVECTORu_SUBu_LZERO, aEXISTSu_DEF, aSUPu_FINITE, aREALu_ARCHu_RDIVu_EQu_0, aNADDu_ADDu_LID, aDIVu_UNIQ, aLEFTu_ANDu_EXISTSu_THM, aARITHu_LTu_conjunct1, aHASu_SIZEu_NUMSEGu_LE, aMATRIXu_VECTORu_MULu_RZERO, aHREALu_ADDu_ACu_conjunct1, aLEu_ADDu_RCANCEL, aREALu_NEGu_GT0, aMATRIXu_SUBu_RZERO, aLENGTHu_APPEND, aREALu_NEGu_LMUL, aREALu_LEu_RDIVu_EQ, aREALu_ARCHu_POW2, aSQRTu_MUL, aARITHu_GEu_conjunct7, aINu_DIFF, aLEu_1u_conjunct0, aLEu_1u_conjunct2, aREALu_SUBu_LNEG, aSUMu_DIFFS, aREALu_MINu_ACIu_conjunct0, aFINITEu_UNIONS, aREALu_SGNu_CASES, aintu_divides, aINTERu_ACIu_conjunct4, aLINEARu_I, aADMISSIBLEu_LAMBDA, aEXISTSu_PAIRu_THM, aNADDu_MULu_WELLDEFu_LEMMA, aMATRIXu_MULu_LID, aHASu_SIZEu_UNIONS, aINTu_ABSu_CASES, aNADDu_LEu_EXISTS, acolumn, aWFu_UREC, aPSUBSET, aNADDu_MUL, aFINITEu_SUBSETu_IMAGEu_IMP, aSUPu_INSERTu_FINITE, aDIMINDEXu_1, aPASTECARTu_INJ, aSETu_RECURSIONu_LEMMA, aMATRIXu_MULu_ASSOC, aNOTu_LE, aINTu_MIN, aVECTORu_MULu_LZERO, aINFu_INSERTu_FINITE, aARITHu_LEu_conjunct3, aEXISTSu_INu_IMAGE, aNADDu_LBOUND, aREALu_LEu_SUPu_FINITE, aREALu_MINu_LT, aLAMBDAu_PAIRu_THM, aHREALu_LEu_ADD, aINTu_MAXu_ACIu_conjunct0, aROWSu_TRANSP, aORTHOGONALu_BASIS, aHASu_SIZEu_CLAUSESu_conjunct0, aSUPPORTu_CLAUSESu_conjunct5, aINSEGu_SUBSET, aVECTORu_ADDu_ACu_conjunct1, aNORMu_LEu_COMPONENTWISE, aRealu_exto_REALu_INV2u_conjunct0, aWFu_REFL, aINTu_ABS, aMATRIXu_CMULu_RZERO, aREALu_POWu_ONE, aGSPEC, aADJOINTu_WORKS, aHREALu_MULu_LID, aFINITEu_SETu_OFu_LIST, aTRIVu_FORALLu_ORu_THM, aSUPPORTu_CLAUSESu_conjunct0, aCARDu_LTu_TRANS, aVECTORu_NEGu_NEG, arealu_powu_conjunct1, aCARDu_LEu_FINITEu_SUBSETS, avec, aASSOC, aMODu_0, aCARDu_ADD2u_ABSORBu_LE, aSUPu_EQ, aREALu_DIVu_EQu_0, aCARDu_LEu_UNIV, aINF, aNADDu_OFu_NUMu_EQ, aARITHu_EQu_conjunct9, aREALu_DIVu_LMUL, aINTu_OFu_NUMu_GE, aREALu_LEu_LMULu_EQ, aDIFFu_INSERT, aCARDu_FINITEu_IMAGE, aCASEWISEu_WORKS, aLEu_MULT2, aREALu_MAXu_ACIu_conjunct2, aDISTu_TRIANGLE, aREALu_LEu_LADDu_IMP, aINSERTu_INSERT, aPSUBSETu_IRREFL, aREALu_LEu_LINV, aNUMSEGu_ADDu_SPLIT, aFORALLu_TRIPLEDu_THM, aCARDu_EQu_TRANS, aBILINEARu_LNEG, aSUMu_UNIONu_LZERO, aCOUNTABLEu_CASES, aREALu_SUPu_EQu_INF, aintu_lt, aIMAGEu_IMPu_INJECTIVEu_GEN, aALLu_conjunct1, aINJECTIVEu_ONu_IMAGE, aREALu_LEu_MULu_EQu_conjunct0, aCARDu_ADDu_LEu_MULu_INFINITE, aINTu_LEu_NEG, aINTu_LEu_MULu_EQu_conjunct0, aintu_powu_th, aintu_negu_th, aREALu_POWu_LT2u_ODDu_EQ, aintu_mod, anumu_RECURSIONu_STD, aSUBu_ADDu_RCANCEL, arealu_negu_th, aIMAGEu_CONST, aREALu_NEGu_MINUS1, aHULLu_ANTIMONO, aLENGTHu_MAP2, anumu_FINITE, aLETu_DEF, aSUMu_MULTICOUNT, aMATRIXu_NEGu_SUB, anaddu_rinv, aFINITEu_RULESu_conjunct1, aREALu_DIVu_POW2u_ALT, aMEMu_APPENDu_DECOMPOSEu_LEFT, aINSERTu_ACu_conjunct0, aMATCHu_SEQPATTERN, aTREALu_ADDu_LID, aMEMu_APPEND, aINTu_SUB, aZBOT, aMATRIXu_VECTORu_MULu_ADDu_RDISTRIB, aINTu_MULu_ACu_conjunct1, aREALu_EQu_LCANCELu_IMP, aREALu_CONVEXu_BOUND2u_LT, aLTu_NZ, aHULLu_IMAGEu_GALOIS, aINTERSu_1, aARITHu_MULTu_conjunct4, aINTu_LEu_SQUARE, aNADDu_ADDu_LCANCEL, aAPPENDu_EQu_NIL, aSUBSETu_TRANS, aREALu_SGNu_EQu_conjunct0, aREALu_RNEGu_UNIQ, aFSTu_DEF, aFINRECu_UNIQUEu_LEMMA, aDIFFu_UNIV, aUNIQUEu_SKOLEMu_THM, aWFu_INTu_MEASUREu_2, aSUMu_POSu_LE, aINDu_SUCu_INJ, aINTu_LTu_MIN, aFACTu_LT, aWOSETu_WELL, adimindex, aDISTu_LEu_0, aITLISTu_conjunct0, aCARDu_EQu_COUNTABLEu_SUBSETSu_REAL, abasis, aCARDu_FUNSPACE, aVSUMu_UNIONSu_NONZERO, aFORALLu_PAIREDu_THM, aREALu_POWu_LE2, aREALu_DIVu_POW2, aLEu_ANTISYM, aTRANSPu_TRANSP, aREALu_POW2u_ABS, aFACTu_LE, aARITHu_ZEROu_conjunct0, aSUMu_RMUL, aIMAGEu_DIFFu_INJ, aDIVMODu_EXISTu_0, aNSUMu_UNIONSu_NONZERO, aINTu_DIVISIONu_0, aWFu_MEASUREu_GEN, aDOTu_RNEG, anaddu_le, aLOWERu_BOUNDu_FINITEu_SETu_REAL, aWFu_POINTWISE, aREALu_SGNu_DIV, aTREALu_MULu_LINV, aCOUNTABLEu_INTER, aREALu_LEu_MUL, aCARDu_IMAGEu_INJu_EQ, aFORALLu_BOOLu_THM, aadmissible, aDISTu_REFL, anumu_gcd, aCARDu_LETu_TOTAL, aDIVu_LE, aCARDu_EQu_COUNTABLE, aSUMu_DELETE, aequ_c, aARITHu_MULTu_conjunct0, aREALu_SGN, aORDINALu_UNIONu_LEMMA, aARITHu_PREu_conjunct1, aDIVISION, aINu_CARDu_ADDu_conjunct0, aREALu_LEu_ADD, aREALu_LEu_POWu_2, aSQRTu_MONOu_LEu_EQ, aREALu_OFu_NUMu_GT, aINu_CARDu_ADDu_conjunct1, aARITHu_EQu_conjunct4, aARITHu_GTu_conjunct0, aSsrnato_lequ_addr, aDIVMODu_UNIQu_LEMMA, aINTu_LETu_ANTISYM, aITERATEu_CLAUSESu_GEN, aAPPROACHABLEu_LTu_LE, aMATRIXu_SUBu_LZERO, aNUMSUMu_INJ, aREALu_POWu_DIV, aintu_minu_th, aSUMu_BOUNDu_LTu_GEN, aNSUMu_NSUMu_RESTRICT, aINTu_LTu_TOTAL, aINTu_ABSu_POW, aORu_DEF, aCONDu_RAND, aREALu_INVu_POW, aITERATEu_RELATED, aNSUMu_CONST, aREALu_LEu_RADD, aHASu_SIZEu_NUMSEGu_1, aSUMu_ABSu_NUMSEG, avectoru_neg, aREALu_POWu_MONO, aNUMu_REPu_RULESu_conjunct1, aARITHu_EXPu_conjunct3, aARITHu_LEu_conjunct0, aVECTORu_SUBu_RZERO, aSUMu_INJECTION, aINTu_POS, aADMISSIBLEu_UNGUARDEDu_PATTERN, aMAPu_conjunct0, aNADDu_UBOUND, aIu_THM, aEXu_conjunct0, aCOUNTABLEu_SUBSET, aVSUMu_IMAGEu_NONZERO, aCARDu_MULu_LTu_INFINITE, aSQRTu_MONOu_LT, aDIVu_1, aEXISTSu_INu_GSPECu_conjunct2, aSUMu_ABSu_LE, aNORMu_BASIS, aSsrboolo_notu_falseu_isu_true, aVSUMu_EQu_GENERALu_INVERSES, aCARDu_NUMSEG, aREALu_POWu_MONOu_LT, aELu_CONS, aSUPPORTu_CLAUSESu_conjunct1, aSUMu_GPu_OFFSET, aINTERu_UNIVu_conjunct1, aINTu_ADDu_SUB2, aPAIRu_EQ, aDELETE, aREALu_BOUNDSu_LE, aSUMu_EQu_NUMSEG, aCARDu_EQu_SUM, aREALu_LTEu_ADD2, alinseg, aFINITEu_DIFF, aMEMu_EXISTSu_EL, aPOSETu_TRANS, aVECTORu_1, aVECTORu_EQu_RDOT, aSUBu_ADDu_LCANCEL, aLEFTu_EXISTSu_IMPu_THM, aCARDu_LEu_ANTISYM, aMODu_LT, aARITHu_ODDu_conjunct1, aINTu_POWu_LTu_1, aVSUMu_SWAP, aNORMu_POWu_2, aREALu_POLYFUNu_EQu_0, aINTu_ADDu_RDISTRIB, aMONOIDALu_MUL, aNSUMu_CLAUSESu_conjunct0, aNUMPAIR, aFORALLu_DIMINDEXu_1, aVSUMu_EQu_0, aEXPu_MONOu_LEu_IMP, aintu_coprime, aNUMSEGu_LREC, aNSUMu_EQu_0, aVECTORu_MULu_ASSOC, aREALu_ABSu_REFL, aALL2u_conjunct1, aCARDu_NUMSEGu_LT, aFROMu_INTERu_NUMSEG, aNUMSUM, aNADDu_MULu_LINVu_LEMMA5, aLEFTu_ORu_FORALLu_THM, aREALu_INVu_1u_LT, aHREALu_MULu_LINV, atailadmissible, aITERATEu_CLOSED, aDISTu_LEu_CASES, aVSUMu_ADD, anaddu_abs, aNORMu_0, aARITHu_EVENu_conjunct1, aCONSu_11, aINTu_SGNu_0, aMATu_COMPONENT, aCONJu_ACIu_conjunct0, aSUMu_CLAUSESu_LEFT, aREALu_LEu_DIV2u_EQ, atrealu_eq, aSUBSETu_RESTRICT, aINTu_ABSu_ABS, aARITHu_MULTu_conjunct8, aODDu_SUB, aCOUNTABLEu_IMAGEu_INJu_EQ, aNSUMu_LEu_NUMSEG, aEVENu_MULT, aNEUTRALu_VECTORu_ADD, aREALu_LEu_ADDL, anumu_MAX, aMULTu_EQu_0, aREALu_POSu_NZ, aLEu_LDIVu_EQ, aLENGTHu_conjunct1, aWOSET, aINTu_POWu_LEu_1, aREALu_LTu_ANTISYM, aADMISSIBLEu_GUARDEDu_PATTERN, aLTEu_ADD2, aINTu_LTu_SUBu_RADD, aVSUMu_DELETEu_CASES, aFINITEu_IMAGEu_INJu_GENERAL, aPOSETu_FLEQ, aMONOu_COND, aFINITEu_IMPu_COUNTABLE, aODDu_DOUBLE, aINu_UNIONS, aANDu_ALL2, aUNIONSu_1, arealu_min, aintu_mulu_th, aALL2u_MAP2, aHREALu_LEu_ADDu_LCANCEL, aintu_add, aARITHu_LEu_conjunct8, aNUMu_OFu_INT, aEQu_ADDu_RCANCEL, aNSUMu_EQu_0u_IFF, aVECTORu_SUBu_ADD2, aINTu_LEu_TOTAL, aSURJECTIVEu_RIGHTu_INVERSE, aCONSTRu_IND, aFINITEu_FUNSPACE, aARITHu_EQu_conjunct1, aINTu_LEu_LMUL, aVECTORu_COMPONENTWISE, aREALu_NEGu_EQu_0, aIMAGEu_DELETEu_INJ, aARITHu_LTu_conjunct2, aSIMPLEu_IMAGEu_GEN, aINTER, aINTu_ADDu_LDISTRIB, aDISJu_ACIu_conjunct3, aREALu_LTu_ADD, aSND, aFINITEu_COLUMNS, aVSUMu_DIFF, aFUNCTIONu_FACTORSu_RIGHTu_GEN, aINTu_MULu_RID, aSUPPORTu_SUBSET, aSUBSETu_EMPTY, aLTu_0, aVSUMu_INCLu_EXCL, aCARDu_UNIONS, aCARDu_LEu_REFL, aINTu_NEGu_MUL2, aINTu_EQu_SUBu_RADD, aFINITEu_INDEXu_INRANGE, aVECTORu_SUBu_RADD, aSUBu_conjunct0, aCARDu_ADDu_FINITEu_EQ, aADJOINTu_UNIQUE, aINTu_LEu_SUBu_RADD, aINTu_LEu_ADDL, aSWAPu_EXISTSu_THM, aLEu_ADDu_LCANCEL, aLEu_C, aSUBSET, aCARDu_COUNTABLEu_CONG, aCARDu_NUMSEGu_LEMMA, aMODu_EXISTS, aCARDu_EQu_REALu_SEQUENCES, aALLu_APPEND, aSUBSETu_INTER, aARITHu_GTu_conjunct9, aSQRTu_POW2, aCARDu_LEu_CARD, aDISJOINTu_EMPTYu_REFL, aMATRIXu_CMULu_ASSOC, aADDu_conjunct0, aSUMu_PAIR, aMATRIXu_WORKS, aADMISSIBLEu_NSUM, aORTHOGONALu_CLAUSESu_conjunct4, aCARDu_NOTu_LT, aSUBSETu_NUMSEG, aINTu_POWu_ONE, aHypermapo_GEu_1, aUPPERu_BOUNDu_FINITEu_SET, aSUMu_POSu_BOUND, aHASu_SIZEu_2, aEUCLIDEANu_SPACEu_INFINITE, aVSUMu_EQu_GENERAL, aEXu_conjunct1, aFINRECu_conjunct0, aIN, aORDINALu_CHAINED, aLTu_LE, aEVENu_ORu_ODD, aARITHu_PREu_conjunct2, aETAu_AX, aINTu_LEu_LNEG, aTREALu_EQu_REFL, aMATRIXu_ADDu_ASSOC, aVSUMu_CASES, aINu_SETu_OFu_LIST, aIMAGEu_CLAUSESu_conjunct0, aLTu_SUC, aINJAu_INJ, aLINEARu_SUB, aUNIONSu_PRED, aPASTECARTu_FSTu_SND, aTREALu_LEu_LADDu_IMP, aEQu_EXP, aEXISTSu_VECTORu_2, aNSUMu_NSUMu_PRODUCT, aNORMu_LEu_0, aLEu_RDIVu_EQ, aREALu_MAXu_SYM, aINTu_POWu_LE2u_ODD, aINTu_LTu_MULu_EQu_conjunct1, aSUBSETu_HULL, aARITHu_EXPu_conjunct6, aMONOIDALu_ADD, aLINEARu_ZERO, aNSUMu_RESTRICTu_SET, aMATRIXu_CMULu_ADDu_LDISTRIB, aMONOu_FORALL, aHREALu_ADDu_ACu_conjunct2, aREALu_ABSu_TRIANGLEu_LT, aNOTu_CLAUSESu_WEAKu_conjunct0, aREALu_POWu_LE2u_ODDu_EQ, aFINITEu_FINITEu_UNIONS, aDIVu_MONOu_LT, aUNIONSu_MONO, aINTu_NOTu_LT, aEXISTSu_ORu_THM, aSQRTu_POWu_2, aARITHu_GEu_conjunct8, aNSUMu_CLAUSESu_RIGHT, aEMPTYu_GSPEC, aANDu_CLAUSES, aADMISSIBLEu_MAP, aINFu_FINITEu_LEMMA, aDISTu_TRIANGLEu_ADD, aREALu_MAXu_MAX, aREALu_INVu_EQu_1, aoneu_tydefu_conjunct1, aNADDu_LEu_RADD, aLOWERu_BOUNDu_FINITEu_SET, aEXISTSu_EX, aCARDu_POWERSET, aFCONSu_conjunct0, aUNIV, aCARDu_MULu_SYM, aoptionu_RECURSION, aLEu_1u_conjunct1, aTARSKIu_SET, aZRECSPACEu_INDUCT, aINTu_LTu_ADD2, aHREALu_MULu_SYM, aCANTORu_THMu_UNIV, aISOu_USAGE, aFNIL, aNORMu_EQu_0u_DOT, aREALu_MULu_RNEG, aFACTu_MONO, aSUMu_OFFSETu_0, aREST, aTRANSPu_MAT, aVSUMu_CLAUSESu_NUMSEGu_conjunct0, aMATRIXu_MULu_RID, aORTHOGONALu_CLAUSESu_conjunct6, aSQRTu_EQu_0, aSUMu_0, aMATRIXu_MULu_DOT, aSELECTu_REFL, aITLISTu_EXTRA, aNSUMu_LE, aINTu_OFu_NUMu_GT, aREALu_NOTu_LE, aARITHu_SUBu_conjunct1, aREALu_SUBu_ADD2, aINSERTu_DEF, aITERATEu_SUPPORT, aREALu_ADDu_SUB2, aDIFFu_EQu_EMPTY, aCONJu_SYM, aSeq2o_HD, aMEMu_conjunct0, aEXISTSu_UNIQUEu_ALT, aWOSETu_TOTALu_LE, aRIGHTu_SUBu_DISTRIB, aSUBSETu_PSUBSETu_TRANS, aINTERSu_INSERT, aREALu_POWu_MUL, aADDu_SUC, aREALu_SOSu_EQu_0, aINTu_MULu_POSu_LE, aCARDu_LEu_RELATIONAL, aVECTORu_2u_conjunct1, aHASu_SIZEu_FUNSPACE, aTREALu_ADDu_WELLDEF, aINTu_MULu_LID, aARITHu_LTu_conjunct6, aCONNECTEDu_REALu_LEMMA, aINTu_LEu_RADD, aCARDu_LEu_TRANS, aFORALLu_UNWINDu_THM2, aREALu_ABSu_LE, aNADDu_MULu_LINVu_LEMMA0, aHULLu_UNIONu_LEFT, aVSUMu_EQu_SUPERSET, aARITHu_GTu_conjunct1, aCANTORu_THM, aisu_int, aREALu_LTu_IMPu_LE, aEXISTSu_INu_INSERT, aALL2u_DEFu_conjunct1, aVECTORu_MULu_RNEG, aINTu_MAXu_MAX, aDOTu_RADD, aDIMINDEXu_UNIV, aINJ, anumu_mod, aWFu_EREC, aINu_SING, aINu_ELIMu_PAIRu_THM, aFORALLu_SIMP, aintu_eq, aTREALu_EQu_AP, aVSUMu_RESTRICTu_SET, aCONDu_DEF, aBIJECTIVEu_INJECTIVEu_SURJECTIVE, aHASu_SIZEu_CROSS, aADMISSIBLEu_IMPu_SUPERADMISSIBLE, aHREALu_ADDu_RID, aCONDu_ELIMu_THM, aMULTu_0, aintu_rep, aLASTu_EL, aNORMu_GEu_SQUARE, aintu_ge, aCARDu_LEu_EMPTY, aINTu_LEu_ANTISYM, aREALu_POWu_ZERO, aPSUBSETu_SUBSETu_TRANS, aSUMu_CLAUSESu_RIGHT, anaddu_inv, aCARDu_LDISTRIB, aHP, aRIGHTu_ORu_EXISTSu_THM, aDISJu_ASSOC, aLINEARu_COMPOSEu_SUB, aFUNCTIONu_FACTORSu_RIGHT, aDISTu_ADD2u_REV, aBIT1u_THM, asumu_RECURSION, aREALu_INFu_UNIQUE, aEXISTSu_TRIPLEDu_THM, aREALu_MINu_ASSOC, aDISJOINT, aNSUMu_BIJECTION, aINTERu_EMPTYu_conjunct1, aIMAGEu_EQu_EMPTY, aMATRIXu_VECTORu_MULu_COMPONENT, aBASISu_EQu_0, aBILINEARu_RZERO, aELu_MAP, aDISTu_ADDBOUND, aDIVu_MONO, aVECTORu_MULu_RCANCEL, aFINITEu_CASES, aREALu_LTEu_ANTISYM, aSUMu_GPu_BASIC, aREALu_LTu_ADDu_SUB, aINTu_ADDu_SUB, aREALu_POLYFUNu_FINITEu_ROOTS, aHULLu_UNIQUE, aBASISu_EXPANSION, aINTu_MINu_MIN, amatrixu_add, aWFu_REC, aZIPu_conjunct1, aBASISu_NONZERO, aREALu_ADD2u_SUB2, aFINITEu_INDEXu_WORKS, aARITHu_GEu_conjunct3, aEQu_TRANS, aRIGHTu_EXISTSu_IMPu_THM, aZRECSPACEu_RULESu_conjunct0, amonoidal, aSUMu_LEu_INCLUDED, aINTu_NEGu_NEG, aINTu_LTEu_ADD2, aMODu_MULTu_MOD2, aMEMu_conjunct1, aREALu_LEu_INF, aFORALLu_FINITEu_INDEX, aINJFu_INJ, aNUMSEGu_REC, aNORMu_BOUNDu_COMPONENTu_LE, aINTu_LTu_ADDu_SUB, aSUBSETu_PRED, aTREALu_LEu_TOTAL, aINTu_OFu_NUMu_MUL, aINTu_LEu_LADDu_IMP, aPAIRWISEu_INSERT, aLEFTu_IMPu_FORALLu_THM, aHULLu_MONO, aPAIRWISEu_conjunct1, aLEu_conjunct0, aLINEARu_CMUL, aMODu_EXPu_MOD, aFUNu_EQu_THM, aINTERu_EMPTYu_conjunct0, aREALu_POS, aMONOIDALu_REALu_MUL, aREALu_LTEu_ADD, aHULLu_P, aDIMINDEXu_UNIQUE, aLEu_1u_conjunct5, aHULLu_REDUNDANT, arealu_sgn, aFINITEu_SUBSETu_IMAGE, aINTu_NEGu_EQu_0, aFINITEu_IMAGEu_INJu_EQ, aARITHu_GEu_conjunct4, aINJECTIVEu_LEFTu_INVERSE, aREALu_SUBu_POLYFUN, aINTERu_ACIu_conjunct3, aNORMu_MUL, aZIPu_conjunct0, aPOWERSETu_CLAUSESu_conjunct0, aNADDu_MULu_LINV, aLEu_SUCu_LT, aINTu_LTu_IMPu_NE, aCARDu_SQUAREu_NUM, agtu_c, aINFINITEu_NONEMPTY, aINFINITEu_CARDu_LE, aPREu_ELIMu_THM, aREALu_SUPu_LEu_SUBSET, aVSUMu_ADDu_GEN, aINTu_LEu_LT, aREALu_LEu_LADD, aINTu_MINu_ACIu_conjunct0, aMATRIXu_NEGu_COMPONENT, aVECTORu_ADDu_ACu_conjunct0, aREALu_SUPu_BOUNDS, aREALu_LTu_LNEG, aLEu_REFL, aCASEWISEu_conjunct1, aWOSETu_POSET, aREALu_MINu_ACIu_conjunct2, aHASu_SIZEu_IMAGEu_INJ, aSUMu_SWAP, aREALu_DIFFSQ, aWOSETu_REFL, aREALu_LTu_REFL, aDIFFu_UNIONSu_NONEMPTY, aREALu_LEu_01, aoneu_RECURSION, aFINITEu_CARTu_UNIV, aLTu_ADD2, aCARDu_LEu_MUL, aNORMu_CAUCHYu_SCHWARZu_ABS, aLTu_MULT, aINTu_MULu_POSu_LT, aREALu_MULu_ACu_conjunct1, aINJECTIVEu_LEFTu_INVERSEu_NONEMPTY, aREALu_LTu_ADDR, aPAIREDu_ETAu_THMu_conjunct0, aNADDu_OFu_NUMu_ADD, aSUBu_PRESUC, aSUMu_DIFF, astringu_INFINITE, aITERATEu_SUPERSET, aEXTENDu_LINSEG, aEXPu_1, aFINITEu_INSERT, aINSEGu_WOSET, aMATRIXu_CMULu_LZERO, aVSUMu_CMULu_NUMSEG, aEXISTSu_UNIQUEu_THM, aUNIONu_ACIu_conjunct2, aCARDu_LEu_ADD, aHASu_SIZEu_PRODUCTu_DEPENDENT, aHREALu_ADDu_LDISTRIB, aARITHu_PREu_conjunct0, aDIVu_MULTu_ADD, aCARDu_ADDu_ASSOC, aINTu_OFu_NUMu_EQ, aMATRIXu_ADDu_LDISTRIB, aWFu_RECu_CASESi_, ahrealu_inv, aINu_DISJOINT, aFORALLu_INu_GSPECu_conjunct2, aIMAGEu_INTERu_INJ, aSUPPORTu_CLAUSESu_conjunct6, aMODu_EQu_0, aintu_abs, aINTu_EQu_ADDu_RCANCEL, aSUMu_3, aARITHu_ODDu_conjunct0, aREALu_SUBu_SUB2, aVSUMu_NORMu_TRIANGLE, aintu_abstr, aCARDu_FINITEu_CONG, aINSERTu_DELETE, aNADDu_LEu_LMUL, aORTHOGONALu_CLAUSESu_conjunct9, aTREALu_ADDu_SYM, aMONOu_OR, aINTu_POWu_LT2, aREALu_LTu_BETWEEN, aREALu_POWu_LTu_1, amulu_c, aEVENu_ADD, aEXPu_MONOu_LTu_IMP, aSUMu_IMAGEu_NONZERO, aFORALLu_VECTORu_2, aMULTu_CLAUSESu_conjunct0, aUNIONu_SUBSET, aMULTu_SYM, aNOTu_EMPTYu_INSERT, aFINRECu_FUN, aREALu_LTu_LCANCELu_IMP, aTREALu_EQu_TRANS, aRECURSIONu_SUPERADMISSIBLE, aREALu_HREALu_LEMMA2, aINTERSu_GSPECu_conjunct2, afrom, aINTu_SUBu_ADD, aREALu_NOTu_EQ, aREALu_ABSu_SUBu_ABS, aORu_EXISTSu_THM, aFORALLu_UNCURRY, aLISTu_OFu_SETu_PROPERTIES, aREALu_LTu_RNEG, aARITHu_EXPu_conjunct4, aSETSPEC, aINDu_SUCu_0, aWO, apairu_cases, aREALu_LEu_MAX, aSQRTu_POSu_LT, aINTu_POWu_NZ, aSUMu_TRIVu_NUMSEG, aREALu_LETu_ADD2, aLTu_conjunct1, aREALu_LTu_SUBu_RADD, aNUMu_OFu_INTu_OFu_NUM, aNSUMu_CLAUSESu_NUMSEGu_conjunct0, aEPSILONu_DELTAu_MINIMAL, aITERATEu_DIFFu_GEN, aNUMSEGu_LT, acong, ahrealu_mulu_th, aBETAu_THM, aINTu_LEu_TRANS, aNADDu_ARCH, aITLISTu_conjunct1, aVSUMu_NORMu_LE, aSUMu_CLAUSESu_conjunct0, arealu_ofu_numu_th, aNSUMu_BOUNDu_LTu_GEN, aNSUMu_EQu_SUPERSET, aVECTORu_SUBu_ADD, aREALu_LTu_LMUL, aREALu_LEu_INV2, aINTERu_ASSOC, aDIVMODu_ELIMu_THM, aFROMu_INTERu_NUMSEGu_GEN, aNORMu_REAL, aREALu_LEu_MULu_EQu_conjunct1, aWOSETu_TRANS, aREALu_ABSu_SIGN2, aLEFTu_SUBu_DISTRIB, aARITHu_EQu_conjunct8, aFORALLu_SUBSETu_IMAGE, aREALu_BOUNDSu_LT, aINTu_LTu_TRANS, aNUMSEGu_LE, aDOTu_BASISu_BASISu_UNEQUAL, aCOMMAu_DEF, aIMAGEu_UNION, aITSET, aUNIONu_IDEMPOT, aREALu_LTu_DIV2u_EQ, aREALu_MULu_SYM, aREALu_LTu_ADD2, aNUMPAIRu_DEST, aEMPTYu_NOTu_UNIV, aintu_pow, aCARDu_LEu_IMAGE, aREALu_LEu_NEGR, aVSUMu_COMBINEu_L, arealu_le, aREALu_ADDu_SUB, aLEu_SQUAREu_REFL, aADDu_ACu_conjunct1, anumu_coprime, aMATRIXu_CMULu_ADDu_RDISTRIB, aARITHu_LEu_conjunct4, aNOTu_FORALLu_THM, aNADDu_ARCHu_MULT, aINTu_LTu_ANTISYM, aSeq2o_MEM, aNORMu_BASISu_1, aREALu_ADDu_SYM, aTREALu_MULu_SYMu_EQ, aPSUBSETu_ALT, aLAST, aHULLSu_EQ, aSUMu_EQu_SUPERSET, aNADDu_LDISTRIB, aPAIRWISEu_EMPTY, aADDu_conjunct1, aSUMu_ADDu_GEN, aDOTu_SYM, aARITHu_GEu_conjunct0, aBASISu_EXPANSIONu_UNIQUE, aINSERT, aFINITEu_REALu_INTERVALu_conjunct0, aSUMu_SING, aVSUMu_SUPERSET, aINu_DELETEu_EQ, aHASu_SIZEu_3, aintu_congruent, aHREALu_COMPLETE, aSQRTu_INJ, aintu_ofu_num, aSQRTu_EVENu_POW2, acartu_tybiju_conjunct1, aABSu_SIMP, aINTu_ABSu_LE, aSINGu_GSPECu_conjunct1, aSUBSETu_REFL, aSUMu_COMBINEu_R, aREALu_HALFu_conjunct1, aITERATEu_BIJECTION, aSYMMETRICu_LINEARu_IMAGE, aSUMu_IMAGEu_LE, aCARDu_LEu_IMAGEu_GEN, abilinear, aHREALu_EQu_ADDu_LCANCEL, aINTu_ABSu_BETWEEN, aITERATEu_OP, aSUPPORTu_CLAUSESu_conjunct2, aIMPu_CONJu_ALT, aINTu_POWu_LE2, aREALu_ABSu_CIRCLE, aMAP2u_DEFu_conjunct0, aINTu_LTu_SUBu_LADD, aINTu_MINu_ACIu_conjunct4, aARITHu_ADDu_conjunct8, aREALu_DIVu_RMUL, aFINITEu_DELETEu_IMP, aALLu_MP, aARITHu_ADDu_conjunct3, aARITHu_ZEROu_conjunct1, aVSUMu_RESTRICT, aHREALu_MULu_RZERO, aINTERSu_2, aHREALu_LEu_EXISTS, aITERATEu_INCLu_EXCL, aINJECTIVEu_MAP, aSQUAREu_BOUNDu_LEMMA, aTRANSPu_MATRIXu_NEG, aTRANSPu_COMPONENT, aVSUMu_CLAUSESu_conjunct1, aITLISTu_APPEND, aDIFFu_DIFF, arealu_powu_conjunct0, aREALu_LEu_RMULu_EQ, aNSUMu_SWAPu_NUMSEG, aFORALLu_INu_CLAUSESu_conjunct0, aREALu_LEu_DOUBLE, anumu_congruent, aVSUMu_PARTIALu_SUC, aMULTu_ACu_conjunct0, aINFINITEu_SUPERSET, aLEFTu_ANDu_FORALLu_THM, aINTu_EXISTSu_POS, aCROSSu_EQu_EMPTY, aREALu_LEu_LSQRT, aINTu_MULu_RNEG, aSETu_OFu_LISTu_EQu_EMPTY, aUNIVu_GSPEC, aREALu_LEu_NEGL, aWFu_URECu_WF, aintu_min, afiniteu_sumu_tybiju_conjunct1, aALLu_FILTER, aNADDu_BOUND, aLEu_ADD2, aSQRTu_UNIQUE, aREALu_POWu_INV, aMULTu_CLAUSESu_conjunct4, aLINEARu_VSUMu_MUL, aINTu_RNEGu_UNIQ, aLEu_MULTu_LCANCEL, aINTu_OFu_NUMu_POW, aALLu_T, aMATRIXu_ADDu_RDISTRIB, aITERATEu_IMAGEu_NONZERO, aUNIONSu_DIFF, aMATRIXu_VECTORu_MULu_TRANSP, aINTu_SUBu_ABS, aLINSEGu_FL, aNADDu_EQu_SYM, aNSUMu_CONSTu_NUMSEG, aIMAGEu_INJECTIVEu_IMAGEu_OFu_SUBSET, aCARDu_INFINITEu_CONG, aFROMu_0, arow, aNADDu_MULu_LINVu_LEMMA8, aREALu_LTu_ADDNEG2, aLINEARu_ID, aREALu_ABSu_MUL, aREALu_ABSu_BETWEEN, aLAMBDAu_UNIQUE, afl, aEMPTYu_UNIONS, aORTHOGONALu_LNEG, aEMPTYu_SUBSET, aBIT1u_DEF, aCARDu_LTu_REFL, aWOSETu_TRANSu_LE, amatrixu_sub, aSURJECTIVEu_FORALLu_THM, apastecart, aintu_sub, aLEu_ADDR, aFINITEu_RULESu_conjunct0, aREALu_LEu_NEG2, aMATRIXu_CMULu_SUBu_LDISTRIB, aSEQu_MONOu_LEMMA, aREALu_LTu_RMUL, aARITHu_MULTu_conjunct5, aDIMINDEXu_GEu_1, aIMAGEu_CLAUSESu_conjunct1, aINTu_LETu_TRANS, aINTERu_SUBSETu_conjunct0, aINTu_EQu_ADDu_LCANCEL, aMATRIXu_SUBu_REFL, aNORMu_BOUNDu_COMPONENTu_LT, avsum, aALL2u_ALL, aWFu_MEASURE, aCARDu_MULu_ASSOC, aPAIRWISEu_IMAGE, aCOUNTABLEu_IMAGEu_INJu_GENERAL, aREALu_SGNu_EQu_conjunct1, aDOTu_LNEG, aEXISTSu_ONEu_REP, aREALu_LTu_NEG2, aREALu_MINu_LE, aSURJECTIVEu_IFFu_INJECTIVE, aINTu_ABSu_SUBu_ABS, aARITHu_SUBu_conjunct6, aREALu_MAXu_ACIu_conjunct0, aSUMu_CLAUSESu_NUMSEGu_conjunct1, aREALu_ABSu_NEG, arealu_sub, aSUMu_ABSu_BOUND, aSETu_CASES, aUNIONu_LEu_ADDu_C, aiterate, aNORMu_SUB, aINFINITE, aINTu_GTu_DISCRETE, aNSUMu_LTu_ALL, aDISTu_TRIANGLEu_ALT, aMAX, aLEu_EXP, aNULLu_conjunct0, aRIGHTu_ORu_DISTRIB, aREALu_LEu_LDIVu_EQ, aTREALu_EQu_IMPu_LE, aCARDu_EQu_0, aARITHu_GTu_conjunct5, aSUCu_INJ, aNUMSUMu_DEST, aIMAGEu_I, aHDu_APPEND, aREALu_LEu_LT, aCARDu_EQu_REALu_IMPu_UNCOUNTABLE, aRATu_LEMMA3, aODDu_conjunct0, aNOTu_EQUALu_SETS, aFORALLu_DOTu_EQu_0u_conjunct1, aCARDu_LEu_ADDL, aGEQu_DEF, aLEFTu_ORu_EXISTSu_THM, aINTERu_ACIu_conjunct0, aNSUMu_BOUND, aFORALLu_POSu_MONO, aMATRIXu_CMULu_COMPONENT, aintu_ofu_numu_th, aCARDu_EQu_BIJECTIONS, aREALu_LTu_TRANS, aINTu_OFu_NUMu_SUB, aCONDu_CLAUSES, aFu_DEF, aINTu_POWu_2, aREALu_ARCHu_INV, aLTu_ADDR, aVECTORu_ADDu_RINV, aINTu_LEu_ADD, aCONJu_ACIu_conjunct2, aARITHu_EQu_conjunct2, aREALu_ADDu_ASSOC, aVECTORu_SUBu_EQ, aSQRTu_LTu_0, aWFu_RECu_WF, aINTu_POWu_MONOu_LT, aCARDu_ADD2u_ABSORBu_LT, aSUBSETu_UNIONu_conjunct0, aVSUMu_UNION, aREALu_EQu_ADDu_RCANCELu_0, aSUMu_BOUND, aREALu_LTu_INV, aNADDu_ADDu_SYM, aHASu_SIZE, aARITHu_EQu_conjunct5, aSUMu_DELETEu_CASES, aORTHOGONALu_0, aINTu_LETu_ADD2, aSURJECTIVEu_ONu_IMAGE, aREALu_LTu_INVu_EQ, aNADDu_INV, aCARDu_ADDu_ABSORBu_LE, aCARDu_FUNSPACEu_CONG, aINTu_SUBu_NEG2, aSUMu_IMAGE, aNADDu_EQu_IMPu_LE, aLTu_MULTu_LCANCEL, aLINEARu_BOUNDED, aSUMu_BIJECTION, aUNIONSu_GSPECu_conjunct2, aPSUBSETu_MEMBER, aPAIR, aMAPu_I, aDISJOINTu_INSERT, aINSERTu_COMM, aREALu_MINu_SYM, aintu_absu_th, aREALu_INVu_NEG, aWFu_RECu_INVARIANT, aDIVu_LT, aAPPENDu_SING, aLINSEGu_INSEG, aISOu_REFL, aINTERu_IDEMPOT, aSWAPu_FORALLu_THM, aPSUBSETu_TRANS, asup, aDISTu_TRIANGLEu_LE, aCARDu_LTEu_TOTAL, aneutral, aNSUMu_UNIONu_RZERO, aGeomdetailo_EQu_EXPAND, aALLu_MAP, aintu_sgnu_th, aEQu_CLAUSES, ahull, aISu_HULL, aSUMu_GP, aREALu_LEu_INFu_FINITE, aDOTu_LSUM, aminimal, aREALu_LEu_NEGTOTAL, aVECTORu_ADDu_LDISTRIB, aFCONSu_UNDO, aMATRIXu_MULu_RZERO, aODDu_MOD, ageu_c, aSUBu_EQu_0, aINTu_EQu_SUBu_LADD, aMATRIXu_ADDu_ACu_conjunct2, aREALu_POWu_EQu_1u_IMP, aNORMu_POSu_LE, aINTu_MULu_ACu_conjunct0, aARITHu_ADDu_conjunct7, aSQRTu_MONOu_LTu_EQ, aINTu_MAXu_ACIu_conjunct3, aREALu_POLYu_NEGu_CLAUSESu_conjunct1, aLTu_LMULT, aCARDu_DELETE, aSUCu_DEF, aLEu_LDIV, aIMAGEu_IMPu_INJECTIVE, aHREALu_ARCH, aCHOOSEu_SUBSETu_BETWEEN, aSUMu_UNION, aFINITEu_HASu_SIZE, aSURJECTIVEu_ONu_RIGHTu_INVERSE, aDELETEu_SUBSET, aDISJu_ACIu_conjunct4, aSUMu_CONSTu_NUMSEG, aVECTORu_CHOOSEu_SIZE, aCARDu_FUNSPACEu_UNIV, aEQu_UNIV, aLTEu_ANTISYM, aTRANSPu_MATRIXu_CMUL, aVSUMu_CLAUSESu_conjunct0, aEXISTSu_INu_GSPECu_conjunct1, aFSTCARTu_PASTECART, aREALu_INFu_BOUNDS, aFINITEu_ROWS, aCARDu_LTu_ADD, aDIFFu_EMPTY, aREALu_POWu_LE, aMEASUREu_LE, aINJN, aNUMSEGu_OFFSETu_IMAGE, aREALu_MULu_LZERO, aREALu_MULu_POSu_LE, aEXISTSu_INu_UNIONS, aLETu_CASES, aCOUNTABLEu_LISTu_GEN, aVECTORu_MULu_COMPONENT, altu_c, aINJPu_INJ, aINTu_NEGu_GT0, aREALu_LTu_LADDu_IMP, aSIMPLEu_IMAGE, aSETu_PROVEu_CASES, aINTu_LTu_ADD1, aNADDu_RDISTRIB, aREALu_COMPLETEu_SOMEPOS, aINTu_LEu_MAX, aNOTu_ALL, aDIVu_MOD, aINTu_NEGu_LT0, aWLOGu_LE, aNADDu_LEu_TOTAL, ahrealu_addu_th, aIMAGEu_UNIONS, aSUMu_CLAUSESu_NUMSEGu_conjunct0, aSURJECTIVEu_IMAGEu_EQ, aNOTu_DEF, aINTu_ABSu_BETWEEN2, aordinal, aNADDu_ADD, aNADDu_MULu_LINVu_LEMMA4, aSUMu_CASESu_1, aORTHOGONALu_CLAUSESu_conjunct1, aZRECSPACEu_RULESu_conjunct1, aLENGTHu_LISTu_OFu_SET, aLINEARu_COMPOSEu_CMUL, aREALu_ADDu_RDISTRIB, aINTu_ABSu_SGN, atrealu_neg, aMAPu_EQu_ALL2, aINTu_SUBu_LZERO, aINTu_LEu_DOUBLE, aCONSTRu_BOT, aSETu_PAIRu_THM, aARITHu_MULTu_conjunct9, aREALu_NEGu_GE0, aBIJECTIVEu_INVERSES, aINTu_ABSu_SIGN, aINTu_GCDu_EXISTS, aBIJECTIONSu_HASu_SIZE, aCARDu_LTu_LE, aFILTERu_MAP, aLTu_ADDu_LCANCEL, aUNIONu_OVERu_INTER, anumu_Axiom, aADMISSIBLEu_SEQPATTERN, aMODu_MULT, aHULLu_INDUCT, aSUPERADMISSIBLEu_COND, aRECURSIONu_CASEWISEu_PAIRWISE, aRATu_LEMMA2, aINTERu_UNIVu_conjunct0, aREALu_MULu_LID, aCOUNTABLEu_RESTRICT, aDISTu_TRIANGLESu_LE, aREALu_HREALu_LEMMA1, aNSUMu_BOUNDu_LT, aARITHu_LTu_conjunct9, aNOTu_CLAUSESu_WEAKu_conjunct1, au_GUARDEDu_PATTERN, aNADDu_INVu_0, aFINRECu_conjunct1, aFORALLu_VECTORu_3, aFINITEu_IMAGEu_EXPAND, aDISTu_SYM, aREALu_LEu_TOTAL, aREALu_EQu_INV2, aCOUNTABLEu_SING, aREALu_SUBu_RNEG, aUNIONSu_2, aINu_NUMSEG, aDISTu_TRIANGLEu_HALFu_L, aINTERu_SUBSETu_conjunct1, aLINEARu_COMPOSEu_VSUM, aHREALu_LEu_ANTISYM, aMKu_RECu_INJ, aFORALLu_INu_IMAGE, aMODu_UNIQ, aINFINITEu_IMAGEu_INJ, aREALu_EQu_LDIVu_EQ, aNUMERAL, aINu_NUMSEGu_0, aPOSETu_RESTRICTEDu_SUBSET, aTREALu_LEu_REFL, aDOTu_LADD, aCARDu_LEu_COUNTABLE, aSUMu_POSu_EQu_0u_NUMSEG, aREALu_POWu_EQu_ABS, aALL2u_conjunct0, ainteger, aITERATEu_DIFF, aGEu_REFL, aHREALu_LEu_MULu_RCANCELu_IMP, aCARDu_LEu_LT, aEVENu_ANDu_ODD, aFILTERu_conjunct1, aARITHu_GTu_conjunct8, aFORALLu_POSu_MONOu_1, aFUNCTIONu_FACTORSu_LEFTu_GEN, aLEFTu_EXISTSu_ANDu_THM, aREALu_INVu_EQu_0, aSNDCARTu_PASTECART, aPu_HULL, aREALu_LTu_INFu_FINITE, aNORMu_LT, asumu_DISTINCT, aREALu_ARCHu_POWu_INV, aNEUTRALu_REALu_ADD, aNOTu_EXISTSu_THM, aREALu_LTu_RSQRT, aCARDu_MULu_ABSORBu_LE, ahrealu_le, aNSUMu_SWAP, aMODu_MODu_REFL, aPASSOCu_DEF, aARITHu_SUCu_conjunct2, anumu_divides, amatrixu_mul, aNOTu_PSUBSETu_EMPTY, aINTu_LEu_MIN, aDISTu_0, aNSUMu_EQu_NUMSEG, aCARDu_MULu_ABSORB, aADDu_ACu_conjunct0, aADDu_SUB2, aARITHu_SUBu_conjunct2, aNORMu_TRIANGLEu_SUB, aNSUMu_EQu_0u_IFFu_NUMSEG, aREALu_LEu_LCANCELu_IMP, aARITHu_EXPu_conjunct5, arealu_mulu_th, aNADDu_MULu_LINVu_LEMMA1, aFORALLu_SUMu_THM, aARITHu_LEu_conjunct9, aSURJECTIVEu_MAP, aINTu_ADDu_ASSOC, aINTu_OFu_NUMu_OFu_INT, aHULLu_UNIONu_RIGHT, aLTu_SUCu_LE, aEXPu_MULT, aMATRIXu_ADDu_SYM, aEXISTSu_PASTECART, aFINITEu_INDUCTu_STRONG, aHULLu_UNION, aNADDu_OFu_NUM, aSUMu_LTu_ALL, aNADDu_LEu_TRANS, aREALu_SUPu_LE, aCARDu_LEu_SUBPOWERSET, aUNIONu_UNIVu_conjunct0, avectoru_add, aLENGTHu_MAP, aREALu_LTu_GT, aVECTORu_MULu_LCANCELu_IMP, aINTu_LEu_01, aDISJu_SYM, aINTu_OFu_NUMu_ADD, aINTu_ABSu_TRIANGLE, aFINITEu_SUMu_IMAGE, aCOMPONENTu_LEu_NORM, aCARDu_ADDu_C, aVSUMu_3, aINTu_ABSu_1, aINu_ELIMu_THMu_conjunct3, aBILINEARu_BOUNDEDu_POS, aCARDu_LTu_IMPu_LE, aARITHu_EXPu_conjunct0, aTREALu_EQu_SYM, aINTERSu_UNION, aELu_conjunct1, aARITHu_ADDu_conjunct0, aREALu_MAXu_ACIu_conjunct1, aCHOICEu_DEF, aWOSETu_TOTAL, aMEMu_EL, aNORMu_TRIANGLE, aINTu_MAX, aREALu_POWu_NEG, aEVENu_conjunct0, aVSUMu_CONSTu_NUMSEG, aFINITEu_REALu_INTERVALu_conjunct4, aSUMu_LMUL, aHASu_SIZEu_NUMSEG, atransp, aCARDu_ADDu_FINITE, aREALu_SGNu_ABS, aSUBSETu_CARDu_EQ, aLASTu_CLAUSESu_conjunct1, aINSEGu_SUBSETu_FL, aSQRTu_1, aREALu_ARCHu_POW, aREALu_SUBu_INV, aORTHOGONALu_CLAUSESu_conjunct5, aFORALLu_1, aREALu_POWu_LE2u_ODD, aFORALLu_PASTECART, aORu_CLAUSES, aMATRIXu_VECTORu_MULu_ASSOC, aMEMu_MAP, aADDu_SUB, aEXISTSu_UNIQUE, aMATRIXu_MULu_RMUL, aFINITEu_NUMSEGu_LE, aREALu_LTu_NEGTOTAL, aintu_max, aREALu_ABSu_SUB, aIMAGEu_SUBSET, aINTu_SUBu_RNEG, aINJECTIVEu_ONu_ALT, aLETu_ADD2, aINTu_ABSu_ZERO, aINTu_SUBu_LNEG, aHypermapo_ZRu_LTu_1, aSUMu_SUBSETu_SIMPLE, aNORMu_CAUCHYu_SCHWARZu_DIV, aONEu_ONE, aSUBSETu_INTERS, aREALu_EQu_IMPu_LE, aINSERTu_INTER, aREALu_DIVu_SQRT, aCARDu_PRODUCT, aARITHu_LTu_conjunct5, aHREALu_MULu_ASSOC, aREALu_LEu_DIV, aREALu_LEu_ANTISYM, aLINSEGu_WOSET, aREALu_SUPu_LEu_FINITE, adestu_intu_rep, aFACTu_NZ, aMATRIXu_VECTORu_MULu_ADDu_LDISTRIB, aINTu_ABSu_SIGN2, aINTu_EQu_IMPu_LE, aREALu_INVu_DIV, aINTERu_UNIONSu_conjunct1, aSUBu_0, aSUMu_EQu_0u_NUMSEG, aINu_INSERT, aHULLu_IMAGE, aSUBu_SUC, aSUPERADMISSIBLEu_MATCHu_UNGUARDEDu_PATTERN, aSUBSETu_INSERTu_DELETE, aCONSTRu_INJ, aSUMu_BOUNDu_LT, aDISTu_RADD, aCONSu_HDu_TL, aFILTERu_APPEND, aREALu_ABSu_NUM, aREALu_LTu_DIV, aPOWERSETu_CLAUSESu_conjunct1, aoneu_tydefu_conjunct0, aDISTu_LMUL, aDIMINDEXu_FINITEu_IMAGE, aVECTORu_ADDu_SYM, amatrixu_inv, aNSUMu_CLAUSESu_conjunct1, aEQu_SYMu_EQ, aSELECTu_AX, aTRANSITIVEu_STEPWISEu_LE, aPAIREDu_EXT, aREALu_EQu_NEG2, aVSUMu_IMAGE, aFORALLu_DEF, aVSUMu_NORMu_ALLSUBSETSu_BOUND, aUNIONu_ACIu_conjunct3, aNSUMu_SINGu_NUMSEG, afiniteu_imageu_tybiju_conjunct1, aNSUMu_DELTA, aARITHu_ADDu_conjunct2, aREALu_LEu_ADD2, aDIVu_REFL, aLINEARu_NEG, aNADDu_NONZERO, aCARDu_EQu_SYM, aSUCu_SUB1, aDIVu_MULT, aALLu_EL, aHASu_SIZEu_IMAGEu_INJu_EQ, aLEu_EXISTS, aINFINITEu_ENUMERATE, aREALu_ABSu_1, aEQu_MULTu_RCANCEL, aFORALLu_DOTu_EQu_0u_conjunct0, aVECTORu_NEGu_0, aHASu_SIZEu_DIFF, aNADDu_EQu_TRANS, aBILINEARu_VSUMu_PARTIALu_SUC, aNADDu_LEu_TOTALu_LEMMA, aINTu_LTu_MUL, aVECTORu_2u_conjunct0, aSUMu_IMAGEu_GEN, aMONOIDALu_VECTORu_ADD, aGT, aLINEARu_BOUNDEDu_POS, aMAPu_REVERSE, aREALu_POWu_POW, aEQu_MULTu_LCANCEL, aINTu_NEGu_0, aLTEu_CASES, aITERATEu_IMAGE, aFINRECu_SUCu_LEMMA, aADDu_ACu_conjunct2, aINTu_SUBu_ADD2, aMiscu_defsu_andu_lemmaso_numu_infinite, aADMISSIBLEu_MATCHu_SEQPATTERN, aCHOOSEu_SUBSETu_STRONG, aNUMu_GCD, aREALu_LSQRTu_LE, aDISJOINTu_NUMSEG, aINTu_LTu_ADDNEG, aREALu_LEu_RMUL, aDESTu_RECu_INJ, aGABSu_DEF, asndcart, aINTu_DIVISION, aINTu_LTu_POW2, aINSERTu_ACu_conjunct1, aBIJ, aDISTu_POSu_LE, aREALu_NEGu_ADD, aTREALu_INVu_0, aARITHu_GTu_conjunct7, aEMPTY, aSQRTu_0, aFORALLu_CURRY, aREALu_SUPu_ASCLOSE, aIu_Ou_ID, aFLu_RESTRICTEDu_SUBSET, aINJECTIVEu_ONu_LEFTu_INVERSE, aINu_UNIV, aARITHu_GEu_conjunct9, aSQUAREu_CONTINUOUS, aNORMu_LTu_SQUARE, aHULLu_INC, aINTERSu_GSPECu_conjunct1, aREALu_OFu_NUMu_SUM, aVECTORu_ADDu_LINV, atrealu_add, aEXISTSu_VECTORu_1, aHD, aBOUNDSu_LINEARu_0, aDIVu_LEu_EXCLUSION, aNADDu_COMPLETE, aHypermapo_THREE, aou_THM, aMAP2u_conjunct1, aintu_tybiju_conjunct0, aEVENu_ODDu_DECOMPOSITION, aSURJECTIVEu_IMAGE, aMONOu_EXISTS, aARITHu_EXPu_conjunct1, aHULLu_REDUNDANTu_EQ, aBOTTOM, aMATRIXu_TRANSPu_MUL, aSUMu_DIFFSu_ALT, aTu_DEF, aINu_INTERS, aMULTu_CLAUSESu_conjunct3, aVECTORu_3u_conjunct1, aREALu_MINu_ACIu_conjunct1, aITERATEu_EXPANDu_CASES, aCARDu_NUMSEGu_LE, aTREALu_ADDu_LINV, aSUMu_SUPERSET, aCARDu_DISJOINTu_UNION, aHASu_SIZEu_NUMSEGu_LT, aPASTECARTu_EQ, aLTu_ADDu_RCANCEL, aREALu_LTEu_TRANS, aNADDu_SUC, aINTu_ABSu_0, arealu_addu_th, aNSUMu_EQ, aNSUMu_POSu_BOUND, aTREALu_MULu_WELLDEFR, aLTu_conjunct0, aTREALu_OFu_NUMu_ADD, aDIVMODu_ELIMu_THMi_, aMULTu_EXP, aREALu_POWu_LT2u_ODD, aAPPENDu_NIL, aFINITEu_INDEXu_INRANGEu_2, aDOTu_NORMu_SUB, aBILINEARu_RSUB, aREFLu_CLAUSE, aMAPu_APPEND, aSUMu_SUBu_NUMSEG, aNUMSEGu_EMPTY, aINTu_MINu_ACIu_conjunct3, aREALu_INVu_1, aARITHu_EQu_conjunct7, arealu_ge, aNOTu_INu_EMPTY, aFINITEu_PRODUCTu_DEPENDENT, aINTu_NOTu_LE, aINTERu_UNIONSu_conjunct0, aNORMu_EQu_0, aORTHOGONALu_SYM, aINJF, aEVENu_DOUBLE, aINTu_ADDu_ACu_conjunct1, aMEMu_FILTER, aMONOu_ALL, aFUNu_INu_IMAGE, aVSUMu_EQu_NUMSEG, aINTERSu_OVERu_UNIONS, aREALu_POWu_1u_LT, aITERATEu_SING, aVECTORu_SUBu_REFL, aPAIRWISEu_SING, aEXISTSu_DIFF, aNEUTRALu_MUL, aVSUMu_COMBINEu_R, aEXISTSu_CURRY, aINTu_OFu_NUMu_MAX, aELu_conjunct0, aPAIRWISEu_MONO, aDISTu_REAL, aCONDu_COMPONENT, aINTu_LEu_RMUL, aVECTORu_NEGu_MINUS1, aREALu_ADDu_RINV, aFINRECu_FUNu_LEMMA, aARITHu_LEu_conjunct5, aNUMSEGu_CLAUSESu_conjunct0, aLINEARu_COMPOSE, aINFu_EQ, aCARDu_UNIONSu_LE, aNADDu_MULu_LINVu_LEMMA7, aPAIRWISEu_conjunct0, aSUMu_2, aINTu_MAXu_SYM, aINTu_OFu_NUMu_LE, aMEMBERu_NOTu_EMPTY, aCASEWISEu_DEFu_conjunct1, aEXu_MEM, aARITHu_SUCu_conjunct0, aREALu_LEu_RNEG, aINTu_EQu_SQUAREu_ABS, aPREu_ELIMu_THMi_, aINTERSu_GSPECu_conjunct0, aREALu_LEu_SQUARE, aREALu_DOWN2, aPAIREDu_ETAu_THMu_conjunct2, aRIGHTu_ADDu_DISTRIB, aREALu_POWu_LE2u_REV, aUNIONSu_INTERS, aREALu_ABSu_DIV, aREALu_LEu_ADDR, aTWO, aWFu_RECu_num, aREALu_SGNu_EQu_conjunct2, aBILINEARu_LMUL, aWOSETu_FLEQ, aLAMBDAu_SKOLEM, aNADDu_MULu_LINVu_LEMMA7a, aWFu_RECu_TAILu_GENERAL, aINJECTIVEu_ALT, aLETu_TRANS, aEXISTSu_THM, aARITHu_SUBu_conjunct3, alambda, aORTHOGONALu_REFL, asumu_CASES, aEMPTYu_DELETE, aINJu_INVERSE2, aVECTORu_EQu_LDOT, aUNIONSu_GSPECu_conjunct0, aDISJOINTu_EMPTY, aMATRIXu_ADDu_COMPONENT, au_SEQPATTERN, anumu_ofu_int, aREALu_POWu_1u_LE, aMATRIXu_SUBu_COMPONENT, aWOSETu_WELLu_CONTRAPOS, aZLu_SUBSETSu_UNIONS, aREALu_NEGu_LT0, aADDu_SUBR, aVECTORu_EQ, aTREALu_ADDu_LDISTRIB, aSUMu_UNIONu_EQ, arealu_gt, aCARDu_IMAGEu_INJ, aRIGHTu_FORALLu_ORu_THM, aINTu_LTu_01, aNADDu_LEu_RMUL, aTRANSPu_MATRIXu_SUB, aHREALu_ADDu_ASSOC, aMATRIXu_ADDu_LID, aHREALu_LEu_ADDu_RCANCEL, aINTu_SGNu_MUL, aINTu_IMAGE, aWF, aTREALu_OFu_NUMu_EQ, aMULTu_ACu_conjunct1, aDISTu_NZ, aMAPu_SNDu_ZIP, aEXTENSION, aINTu_DIVMODu_EXISTu_0, aPREu_conjunct1, aREALu_ABSu_CASES, aNADDu_OFu_NUMu_WELLDEF, aDISTu_RADDu_0, aMONOIDALu_REALu_ADD, aREALu_COMPLETE, aNUMSEGu_SING, aCARDu_LEu_INJ, aVSUMu_PAIR, aVECTORu_NEGu_SUB, aSUBu_ELIMu_THM, aARITHu_PREu_conjunct3, aLINEARu_VSUM, aLETu_ENDu_DEF, aINFu_FINITE, aCARDu_UNIONu_GEN, aCOLUMNu_TRANSP, aADMISSIBLEu_MATCH, atrealu_le, aREALu_POWu_1, aDIVu_MULT2, aREALu_LEu_LNEG, aFORALLu_INu_INSERT, aINTu_LTu_ADD, aVSUMu_2, aNSUMu_SUBSET, aINTu_ABSu_NZ, aUNIONu_EMPTYu_conjunct1, aORDINALu_UP, aINu_CROSS, aFORALLu_VECTORu_1, aWFu_FALSE, aSUMu_MULTICOUNTu_GEN, aRATu_LEMMA4, aSNDu_DEF, aINTu_ADDu_SYM, aINTu_MAXu_LT, ahrealu_ofu_numu_th, aEXISTSu_UNIQUEu_DEF, aCARDu_LEu_ADDR, amatrix, aARITHu_GEu_conjunct5, aFACTu_conjunct0, aLEu_1u_conjunct4, aREALu_ADDu_ACu_conjunct0, aSUBSETu_DELETE, aNADDu_MULu_SYM, aNOTu_SUC, aHULLu_EQ, arealu_mod, aINTu_LEu_ADD2, aSUPu_SING, aSURJECTIVEu_EXISTSu_THM, aSUMu_SWAPu_NUMSEG, aARITHu_LEu_conjunct6, aCOUNTABLE, aCARDu_CARTu_UNIV, aHASu_SIZEu_BOOL, aTRANSITIVEu_STEPWISEu_LT, amku_pairu_def, aINTu_LTu_LE, aNSUMu_UNIONu_NONZERO, acharu_INDUCT, aREALu_MULu_RINV, aSUBu_REFL, aREPLICATEu_conjunct1, aCARDu_LTu_CONG, aNADDu_ARCHu_LEMMA, aboolu_RECURSION, aLINEARu_0, aREALu_MULu_ACu_conjunct2, aHULLu_IMAGEu_SUBSET, aMONOu_IMP, aARITHu_EVENu_conjunct0, aHREALu_ADDu_RDISTRIB, aITERATEu_DELETE, aDIVMODu_UNIQ, aINTu_MINu_ASSOC, aNSUMu_CLAUSESu_NUMSEGu_conjunct1, aMODu_MULTu_LMOD, aUNIONu_INSEG, aINTu_POWu_LT, aREALu_POWu_EQu_EQ, aMATRIXu_ADDu_ACu_conjunct1, aCARDu_CLAUSESu_conjunct1, aREALu_SUBu_NEG2, aVECTORu_MULu_RCANCELu_IMP, aMONOIDALu_AC, aEQu_IMPu_LE, aANDu_FORALLu_THM, aNORMu_LTu_SQUAREu_ALT, aSQRTu_INV, aINTu_POWu_NEG, aIMPu_IMP, aINTu_LTu_RADD, aSUMu_COMBINEu_L, aADJOINTu_ADJOINT, aLAMBDAu_PAIR, aVSUMu_SWAPu_NUMSEG, aUNCOUNTABLEu_REAL, aleu_c, aODDu_conjunct1, aDISTu_RMUL, aARITHu_ODDu_conjunct3, aINSERTu_UNIV, aINTu_LTEu_ADD, aNSUMu_LMUL, aDIFFu_UNIONS, aITERATEu_DELTA, au_FUNCTION, aREALu_CONVEXu_BOUNDu_LT, aNORMu_NEG, aINTu_LETu_ADD, aORTHOGONALu_CLAUSESu_conjunct8, aSUMu_BOUNDu_GEN, aTREALu_MULu_WELLDEF, aDOTu_RMUL, aMODu_MOD, aSUPu_UNIQUEu_FINITE, aCARDu_EQu_FINITEu_SUBSETS, aCURRYu_DEF, aKLu_POSETu_LEMMA, aTRANSPu_MATRIXu_ADD, aVSUMu_PAIRu_0, aINTu_NEGu_RMUL, aCONDu_ID, aCARDu_IMAGEu_EQu_INJ, aINTu_SGNu_EQu_conjunct0, aNSUMu_RESTRICT, aUNIVu_SUBSET, aINTu_ABSu_BETWEEN1, aCARDu_ADDu_ABSORB, aINTu_POWu_conjunct1, aLENGTHu_EQu_NIL, aNADDu_LEu_WELLDEF, aSUBu_ADD, aREALu_ABSu_BOUNDS, aADMISSIBLEu_RAND, aNADDu_ADDu_WELLDEF, aDOTu_BASISu_BASIS, aCARDu_MULu_LTu_LEMMA, aMODu_NSUMu_MODu_NUMSEG, aINTu_SGN, aREALu_LEu_SQUAREu_ABS, aREALu_ABSu_INFu_LE, aEQu_REFL, aINJP, aEVENu_MOD, aZIPu_DEFu_conjunct1, aEXPu_MONOu_LT, aSUMu_CLOSED, aMULu_Cu_UNIV, aVSUM, aHREALu_OFu_NUMu_EQ, aZLu_SUBSETS, aINFu_SING, arealu_INFINITE, aCARDu_PSUBSET, aMODu_LE, aCARDu_SUBSETu_LE, aHREALu_ADDu_LCANCEL, aREALu_HALFu_conjunct2, aINTu_POWu_EQu_ABS, aCARDu_LEu_SQUARE, aFORALLu_INu_GSPECu_conjunct0, aINTu_ABSu_MUL, aARITHu_LTu_conjunct8, aDISTu_LZERO, aREALu_EQu_MULu_RCANCEL, aSUMu_ZEROu_EXISTS, aVSUMu_DELTA, ahrealu_invu_th, aINTu_MINu_LT, aRATu_LEMMA1, aDISJOINTu_DELETEu_SYM, aIu_DEF, aMATRIXu_VECTORu_MULu_SUBu_RDISTRIB, aMATRIXu_NEGu_MINUS1, aREALu_ENTIRE, aVSUMu_OFFSET, aREALu_DIVu_REFL, aREALu_OFu_NUMu_MAX, au_MATCH, anaddu_ofu_num, aCARDu_LTu_FINITEu_INFINITE, ainseg, aDOTu_2, aSQRTu_WORKS, aSUMu_CLAUSESu_conjunct1, aHREALu_OFu_NUMu_ADD, aDOTu_POSu_LT, aNSUMu_MULTICOUNT, aCARDu_LEu_FINITE, aZEROu_DEF, aINTERu_COMM, aFORALLu_ALL, aREALu_MULu_LINVu_UNIQ, aREALu_ABSu_SGN, aMATRIXu_SUBu_LDISTRIB, aNADDu_MULTIPLICATIVE, aREALu_LEu_BETWEEN, aSUBSETu_UNIV, aINTu_ABSu_REFL, aDELETEu_NONu_ELEMENT, aINTu_POWu_1u_LE, afiniteu_sumu_tybiju_conjunct0, atrealu_inv, aORDINALu_CHAINEDu_LEMMA, aREALu_POWu_LBOUND, aWOSETu_TRANSu_LESS, aINu_ELIMu_THMu_conjunct2, aSUPERADMISSIBLEu_TAIL, aLTu_TRANS, aNUMSEGu_RREC, aSUMu_GPu_MULTIPLIED, aTRANSITIVEu_STEPWISEu_LTu_EQ, aADDu_EQu_0, aNSUMu_UNION, aINTERSu_UNIONS, aZL, aREALu_EQu_SQUAREu_ABS, aCARDu_BOOL, aVSUMu_CLAUSESu_RIGHT, aRIGHTu_IMPu_EXISTSu_THM, aINJA, aVSUMu_IMAGEu_GEN, acolumns, aCONDu_ABS, aINJECTIVEu_IMAGE, aNADDu_ADDITIVE, aEMPTYu_UNION, aLTu_REFL, aFINITEu_RESTRICT, avectoru_matrixu_mul, aINTu_POWu_1u_LT, aORTHOGONALu_CLAUSESu_conjunct2, aVSUMu_UNIONu_RZERO, aARITHu_EXPu_conjunct9, aINTu_MINu_LE, aDISTu_ELIMu_THM, aNADDu_ADDu_ASSOC, asumu_INJECTIVEu_conjunct1, aAPPENDu_conjunct0, aARITHu_SUCu_conjunct1, aARITHu_LEu_conjunct2, amat, aNADDu_EQu_REFL, aSUMu_ADD, aVECTORu_SUB, aREALu_NEGu_MUL2, aNEUTRALu_REALu_MUL, aSUP, aBOUNDSu_LINEAR, aREALu_LEu_MIN, aFINITEu_IMAGEu_IMAGE, arealu_leu_th, aposet, aLEu_ADD, aFINITEu_UNIONu_IMP, aOUTL, aLEFTu_FORALLu_IMPu_THM, aMIN, aZRECSPACEu_CASES, aARITHu_GEu_conjunct1, aINTu_ENTIRE, aFST, aCARDu_CROSS, aREALu_MAXu_SUP, aCOUNTABLEu_ASu_IMAGEu_SUBSETu_EQ, aVSUMu_INJECTION, aINTu_MAXu_ACIu_conjunct2, aINTu_NEGu_LMUL, aALL2u_MAP, aBASISu_COMPONENT, aSETu_OFu_LISTu_OFu_SET, aSUMu_UNIONu_RZERO, amatrixu_cmul, aREALu_ARCHu_SIMPLE, aEXISTSu_BOOLu_THM, aLEFTu_FORALLu_ORu_THM, aNSUMu_LT, aREALu_LTu_MUL2, aORDINALu_UNION, aCOUNTABLEu_CROSS, aITLIST2u_conjunct0, aDOTu_LSUB, aHULLu_HULL, aDELETEu_INSERT, aSsrnato_leqE, aMODu_MULTu_RMOD, aREALu_SUBu_TRIANGLE, aNSUMu_INCLu_EXCL, aREALu_POWu_2, aREALu_ADDu_LID, aorthogonal, aCOUNTABLEu_UNIONu_IMP, aTREALu_LEu_ANTISYM, aREVERSEu_APPEND, aREALu_ADDu_LDISTRIB, aINTu_LNEGu_UNIQ, aINTu_MAXu_MIN, aREALu_EQu_ADDu_LCANCEL, aLTu_ANTISYM, anumu_WF, aINFINITEu_DIFFu_FINITE, aBILINEARu_RNEG, aITERATEu_CLAUSESu_NUMSEG, aCARDu_IMAGEu_LE, aUPPERu_BOUNDu_FINITEu_SETu_REAL, aREALu_LTu_ADDNEG, aREALu_MULu_2, aREALu_LTu_MULu_EQu_conjunct0, aDECOMPOSITION, apairu_RECURSION, aADD1, aITSETu_EQ, aREALu_LTu_MAX, aSUBSETu_UNIONu_ABSORPTION, aEQu_SYM, aEXPu_MONOu_LE, aREALu_LTu_IMPu_NE, aINTu_OFu_NUMu_SUC, aREALu_MAXu_ACIu_conjunct4, aREALu_ABSu_POW, aINTu_LTu_RMULu_EQ, aMODu_MULT2, aVSUMu_DIFFSu_ALT, aCONJu_ACIu_conjunct1, aINTu_DIVMODu_UNIQ, aEQu_IMP, aINTu_LTu_NEG, aDOTu_NORM, aREALu_SGNu_INV, aNSUMu_UNIONu_EQ, aDIVISIONu_SIMPu_conjunct0, anumu_FINITEu_AVOID, aSUMu_CONST, aLASTu_CLAUSESu_conjunct0, aintu_subu_th, aCOUNTABLEu_ALT, aDELETEu_INTER, aBUTLASTu_conjunct1, aINTu_ABSu_NUM, aCARDu_EQu_FINITE, aBILINEARu_LSUB, amatrixu_vectoru_mul, adist, aIMPu_CONJ, arows, aNSUMu_ADD, aNUMPAIRu_INJ, aCARDu_LTEu_TRANS, aARITHu_GTu_conjunct2, aEVENu_SUB, aBILINEARu_VSUMu_PARTIALu_PRE, aVSUMu_SINGu_NUMSEG, aMEMu_APPENDu_DECOMPOSE, aMAPu_o, aARITHu_MULTu_conjunct6, aVSUMu_EQ, aREALu_LTu_SUBu_LADD, aREALu_LTEu_TOTAL, aREALu_ABSu_ABS, aHREALu_ADDu_SYM, aHULLu_SUBSET, aSUPPORTu_SUPPORT, aINTu_NEGu_SUB, aCARDu_LEu_SUBSET, aMULTu_ACu_conjunct2, aTREALu_ADDu_WELLDEFR, aSINGu_GSPECu_conjunct0, aEXISTSu_SUMu_THM, aSUMu_SUPPORT, aINTu_EQu_ADDu_RCANCELu_0, aREALu_LTu_ADDL, anumu_INFINITE, aARITHu_ADDu_conjunct6, aOEP, aLINEARu_INJECTIVEu_0, aou_ASSOC, aFINITEu_IMAGE, aINTu_MULu_RZERO, aSUMu_PARTIALu_PRE, aREALu_EQu_MULu_LCANCEL, aINu_ELIMu_PASTECARTu_THM, aREALu_SUBu_ABS, aHASu_SIZEu_FUNSPACEu_UNIV, aFINITEu_INDEXu_INJ, aCOUNTABLEu_IMAGEu_INJ, aCONDu_RATOR, aNULLu_conjunct1, aINTu_SUBu_SUB2, aSUMu_SUB, aMAPu_ID, avector, aARITHu_LTu_conjunct4, aIMAGE, aSUMu_ADDu_NUMSEG, aCOLUMNSu_TRANSP, aEXISTSu_NOTu_THM, aoneu_DEF, aINTu_POSu_NZ, aNSUMu_ADDu_NUMSEG, aDIVu_MULu_LE, aITERATEu_ITERATEu_PRODUCT, aCARDu_MUL2u_ABSORBu_LE, aDIVu_DIV, aDIVu_EQu_0, aREALu_POLYFUNu_ROOTBOUND, aREALu_WLOGu_LT, aFINITEu_IMAGEu_INJ, aFINRECu_1u_LEMMA, aVECTORu_MULu_RZERO, aHASu_SIZEu_CARTu_UNIV, aFINITEu_REALu_INTERVALu_conjunct5, aSUMu_GROUP, aTL, aINTu_LTu_NEGTOTAL, aCARDu_SETu_OFu_LISTu_LE, aBIT0u_DEFu_conjunct1, aREVERSEu_conjunct0, aREALu_MULu_ASSOC, aREALu_ADDu_RID, aDOTu_LMUL, aTREALu_OFu_NUMu_MUL, aADMISSIBLEu_COND, aLTu_EXISTS, aPSUBSETu_UNIV, aMAPu_EQu_DEGEN, aRECURSIONu_CASEWISE, aFINITEu_SUPPORT, aEXISTSu_UNCURRY, aINTu_ABSu_STILLNZ, aNUMu_REPu_INDUCT, aREALu_INVu_1u_LE, aSUPu_FINITEu_LEMMA, aLENGTHu_TL, aREALu_DOWN, aBASISu_NE, aINTu_LTEu_TOTAL, aITERATEu_CLAUSES, aNSUMu_RMUL, aCARDu_EQu_LIST, aEMPTYu_DIFF, aADMISSIBLEu_CONST, aMATRIXu_EQ, aCOUNTABLEu_UNION, aNADDu_OFu_NUMu_LE, aIMPu_CLAUSES, aVECTORu_SUBu_LDISTRIB, aINTERu_ACIu_conjunct2, aDOTu_SQUAREu_NORM, aSUPPORTu_DELTA, aHASu_SIZEu_POWERSET, aREALu_SUBu_LE, aINTu_MAXu_LE, aSsrnato_lequ_eqVlt, aREALu_OFu_NUMu_GE, aNADDu_CAUCHY, aBILINEARu_BOUNDED, aREALu_ADDu_ACu_conjunct1, aDIMINDEXu_3, aINTu_SUBu_LT, aNORMu_EQu_0u_IMP, aHREALu_LEu_TRANS, aLTu_ADD, aFLu_RESTRICT, aHASu_SIZEu_0, aNADDu_MULu_LID, aCARDu_ADDu_SYM, aDISTu_TRIANGLEu_ADDu_HALF, aCARDu_LTu_CARD, asetu_ofu_listu_conjunct1, aUNCURRYu_DEF, aCARDu_FUNSPACEu_CURRY, aVSUMu_VMUL, asumu_INDUCT, aINTu_ARCH, aUNIONu_UNIVu_conjunct1, aNOTu_LT, aDISJu_ACIu_conjunct0, aFINITEu_FUNSPACEu_UNIV, aFORALLu_2, aARITHu_GTu_conjunct6, aREALu_EQu_SUBu_RADD, anaddu_mul, aWFu_num, aSUMu_UNIONSu_NONZERO, aCARDu_EQu_IMPu_LE, aUNIVu_NOTu_EMPTY, aCARDu_NOTu_LE, aINTu_SGNu_EQu_conjunct1, aREALu_SUBu_REFL, aREALu_ABSu_BETWEEN1, aFORALLu_INu_CLAUSESu_conjunct1, aINTu_ADD2u_SUB2, aDISTu_LADD, aLEFTu_ORu_DISTRIB, aRIGHTu_EXISTSu_ANDu_THM, aUNIONu_ASSOC, aCASEWISEu_DEFu_conjunct0, aDISTu_LADDu_0, aDOTu_EQu_0, aHASu_SIZEu_SETu_OFu_LIST, aMULTu_DIVu_LE, aCARDu_ADDu_CONG, aREALu_SGNu_MUL, aINTu_POWu_1, aaddu_c, aINTu_ADDu_RID, aREALu_OFu_NUMu_MUL, aCARDu_EQu_IMAGE, aoneu_axiom, aREALu_ABSu_POS, aDIMINDEXu_NONZERO, aINTu_SUBu_0, aTRIANGLEu_LEMMA, aSUPPORTu_CLAUSESu_conjunct3, aCARDu_MULu_CONG, aCOUNTABLEu_ASu_INJECTIVEu_IMAGE, aREALu_SUBu_POWu_L1, aREALu_NOTu_LT, aWFu_RECu_TAIL, aSQRTu_POSu_LE, aEVENu_conjunct1, aDIVu_0, aNUMPAIRu_INJu_LEMMA, aCHOICE, aBIJECTIONSu_CARDu_EQ, aVECTORu_EQu_ADDR, au_FALSITYu_, aDECIMAL, aWFu_LEXu_DEPENDENT, aCARDu_EQu_CONG, aUNIONSu_GSPECu_conjunct1, aUNIONSu_INSERT, aMATRIXu_SUBu_RDISTRIB, aINTu_SUBu_LE, aEQu_Cu_BIJECTIONS, aINTu_ABSu_CIRCLE, aCARDu_EQu_NSUM, aNADDu_INVu_WELLDEF, aVECTORu_3u_conjunct2, aSELECTu_UNIQUE, aNADDu_LEu_0, aNADDu_MULu_LINVu_LEMMA3, aINTu_ABSu_BOUND, aREALu_INFu_LTu_FINITE, aINTu_ADDu_ACu_conjunct2, aPOSETu_REFL, aINTu_SUBu_RZERO, aSUPERADMISSIBLEu_MATCHu_GUARDEDu_PATTERN, aintu_addu_th, aANDu_ALL, aDOTu_RZERO, aTRIVu_ORu_FORALLu_THM, aMATRIXu_VECTORu_MULu_LZERO, aRIGHTu_FORALLu_IMPu_THM, aSUBu_ELIMu_THMi_, aARITHu_EVENu_conjunct3, aFINRECu_EXISTSu_LEMMA, aREALu_LEu_RCANCELu_IMP, aSsrnato_eqnu_leq, aCARD, aREALu_POLYFUNu_EQu_CONST, aARITHu_MULTu_conjunct2, aNORMu_LEu_SQUARE, aINTu_BOUNDSu_LE, aMATRIXu_ADDu_LNEG, aINTu_SGNu_ABS, aREALu_ABSu_TRIANGLEu_LE, aintu_gt, aREALu_LTu_RINV, aADJOINTu_CLAUSES, aCONSTRu_REC, aTRUTH, aEXPu_MONOu_EQ, aARITHu_SUBu_conjunct7, aINTu_LTu_ADDR, aREALu_LTu_RCANCELu_IMP, afiniteu_imageu_tybiju_conjunct0, aSUMu_RESTRICTu_SET, aREALu_SGNu_NEG, aEXISTSu_INu_CLAUSESu_conjunct1, aHULLu_UNIONu_SUBSET, aSUMu_1, aREALu_OFu_NUMu_LE, aSUBSETu_ANTISYM, aCOUNTABLEu_PRODUCTu_DEPENDENT, aVSUMu_1, aFUNCTIONu_FACTORSu_LEFT, aadjoint, aWFu_RECu_CASES, aCOUNTABLEu_CART, aARITHu_EVENu_conjunct2, aINTu_POW2u_ABS, aNSUMu_SUBSETu_SIMPLE, aNADDu_MULu_LINVu_LEMMA6, aCONTRAPOSu_THM, aVSUMu_CLAUSESu_LEFT, aCARDu_UNIONu_EQ, aNUMSEGu_COMBINEu_R, aEXu_IMP, aREALu_NEGNEG, aARITHu_LTu_conjunct0, aSUMu_LEu_NUMSEG, aSUMu_OFFSET, aPOSETu_ANTISYM, aINTu_LT, aDOTu_RSUM, aREALu_EQu_RDIVu_EQ, aFINITEu_RECURSION, aCARDu_SUBSET, aSINGu_SUBSET, asupport, aVECTORu_3u_conjunct0, aREALu_SUBu_LZERO, aEXTENDu_FL, aTRIVu_ANDu_EXISTSu_THM, aTREALu_NEGu_WELLDEF, aREALu_SUBu_POLYFUNu_ALT, aNUMSEGu_COMBINEu_L, asqrt, aCARDu_EQu_LISTu_GEN, anumseg, aNSUMu_EQu_GENERAL, aZIPu_DEFu_conjunct0, aMAP2u_conjunct0, aVSUMu_ADDu_SPLIT, aFORALLu_ANDu_THM, aMODu_EQ, aLTu_CASES, aLINEARu_COMPONENTWISE, aREALu_SUPu_LTu_FINITE, aVSUMu_PARTIALu_PRE, aINSERTu_SUBSET, aTRANSITIVEu_STEPWISEu_LEu_EQ, aINSERTu_UNION, aVECTORu_ADDu_LID, aFINITEu_INDUCTu_DELETE, aREALu_EQu_RCANCELu_IMP, aREALu_POWu_LEu_1, aINTu_POWu_EQu_0, aMATRIXu_ADDu_RNEG, aINTu_GT, aARITHu_EXPu_conjunct2, aHULLu_MINIMAL, aBOUNDSu_NOTZERO, aEXPu_conjunct1, aSUBSETu_IMAGE, aCOUNTABLEu_INSERT, aREALu_OFu_NUMu_LT, aREALu_ABSu_INV, aITERATEu_UNIONu_NONZERO, aINu_INTER, aNORMu_EQu_SQUARE, aHASu_SIZEu_PRODUCT, aNOTu_IMP, aVSUMu_SUB, aREALu_MULu_LNEG, aINu_UNION, ahrealu_mul, aGEu_C, aNSUMu_UNIONu_LZERO, aNOTu_EX, aCARDu_RDISTRIB, aADDu_SYM, aREALu_LTu_LDIVu_EQ, aNADDu_LEu_REFL, aVECTORu_ONE, aINTERu_OVERu_UNION, aONTO, aFORALLu_REALu_ONE, aINTu_ABSu_MULu_1, aEXu_MAP, aSUBu_conjunct1, aINTu_NEGu_LE0, aEVENu_EXISTS, aUNIONu_COMM, aVSUMu_UNIONu_NONZERO, aFINITEu_CART, aFINITEu_CARDu_LT, aREALu_LEu_INV, aINTu_ADDu_ACu_conjunct0, aNADDu_LEu_LADD, aVSUMu_GROUP, aREALu_LTu_MUL, aCONJu_ACIu_conjunct4, aREALu_MAXu_MIN, aGE, aBIT0u_DEFu_conjunct0, aUNWINDu_THM2, aINTu_GE, aNORMu_GTu_SQUARE, aPREu_conjunct0, aINTu_ABSu_NEG, aTRANSPu_EQ, aVSUMu_CASESu_1, aIMAGEu_o, aREALu_NEGu_LE0, aINTu_SUBu_RDISTRIB, aLEu_TRANS, aREALu_OFu_NUMu_SUMu_NUMSEG, aITERATEu_EQ, aboolu_INDUCT, aUNIONu_ACIu_conjunct4, aDISJOINTu_UNION, aINTu_FORALLu_POS, aINTu_LTu_GT, acharu_RECURSION, aFINITEu_INTER, ainvertible, arealu_div, aDIVu_ADDu_MOD, aFORALLu_PAIRu_THM, aVSUMu_OFFSETu_0, aHREALu_ADDu_ACu_conjunct0, aEVENu_EXISTSu_LEMMA, aREALu_OFu_NUMu_EQ, aLEu_1u_conjunct3, aREALu_LETu_TRANS, aINu_ELIMu_THMu_conjunct4, aTREALu_LEu_MUL, aEXISTSu_PAIREDu_THM, aSUMu_RESTRICT, asumu_INJECTIVEu_conjunct0, asum, aDISTu_MUL, aVSUMu_RMUL, aINTu_EQu_ADDu_LCANCELu_0, aARITHu_GEu_conjunct6, aVECTORu_MULu_LID, aADDu_SUBR2, aMATRIXu_VECTORu_MULu_SUBu_LDISTRIB, aREALu_NEGu_SUB, aEXPu_ONE, aCHOOSEu_SUBSET, aALLu_IMP, aHASu_SIZEu_CLAUSESu_conjunct1, aREALu_MULu_RZERO, aITLIST2u_DEFu_conjunct1, aELu_TL, aINTu_LTu_MULu_EQu_conjunct0, aSQRTu_DIV, aTREALu_OFu_NUMu_LE, aINTu_EQu_MULu_RCANCEL, aSUMu_DELTA, aLTu_IMPu_LE, aVSUMu_LMUL, aCOUNTABLEu_DELETE, aINTERSu_0, aREALu_MINu_ACIu_conjunct4, aDISJu_ACIu_conjunct1, aVECTORu_SUBu_RDISTRIB, aVECTORu_ADDu_ACu_conjunct2, aSUMu_CASES, aREALu_OFu_NUMu_SUB, aRIGHTu_ORu_FORALLu_THM, aNSUMu_BOUNDu_LTu_ALL, aDISJOINTu_SYM, aREALu_LEu_POW2, aREALu_POWu_SUB, aREALu_MINu_INF, aBIT0u_THM, aCARDu_SQUAREu_INFINITE, aLAMBDAu_BETA, aCARDu_UNION, aADMISSIBLEu_COMB, anumu_WOP, aINTu_EQu_MULu_LCANCEL, aFINITEu_INDEXu_NUMSEG, aREALu_SUBu_LT, aSURJECTIVEu_IFFu_INJECTIVEu_GEN, aREALu_LTu_SQUARE, aMULTu_2, aRIGHTu_ANDu_EXISTSu_THM, aONE, aMATRIXu_NEGu_ADD, aARITHu_MULTu_conjunct1, aLETu_ANTISYM, aUNIONSu_IMAGE, aCROSS, aINTu_LEu_RNEG, aALL2u_conjunct3, aCARDu_LEu_INFINITE, aLAMBDAu_ETA, aLEu_MULTu_RCANCEL, aFINITEu_NUMSEG, aVSUMu_DIFFS, aSUMu_INCLu_EXCL, aNORMu_EQ, aHASu_SIZEu_CARD, anaddu_eq, aMODu_MODu_EXPu_MIN, aSUBSETu_ANTISYMu_EQ, aREALu_ABSu_BOUND, aEXISTSu_FINITEu_SUBSETu_IMAGE, aless, aINTERSu_IMAGE, aREALu_NEGu_RMUL, aLEu_SUC, arealu_inv, aINTu_LTEu_ANTISYM, aCOUNTABLEu_ASu_IMAGE, aREALu_POLYu_CLAUSESu_conjunct8, aNSUMu_IMAGEu_GEN, aVECTORu_MULu_LNEG, aEXPu_LTu_0, arealu_neg, aintu_maxu_th, ahrealu_leu_th, aNSUMu_DIFF, aREALu_SUBu_POWu_R1, aRATu_LEMMA5, aINTu_MULu_SYM, alistu_INDUCT, aREALu_LEu_REFL, aINTu_LEu_MUL, aMULTu_CLAUSESu_conjunct2, aORTHOGONALu_CLAUSESu_conjunct7, aADDu_ASSOC, aUNIONu_EMPTYu_conjunct0, aINTu_NEGu_GE0, aEXPu_ZERO, aINTu_LEu_MULu_EQu_conjunct1, aEVENu_EXP, aADMISSIBLEu_SUM, aFACTu_conjunct1, aINTu_SUBu_LDISTRIB, aINTu_POWu_ADD, aLENGTHu_conjunct0, aTacticsu_jordano_unifyu_existsu_tacu_example, aINu_IMAGE, aSUPPORTu_EMPTY, aDIFFu_INTERS, aHREALu_LEu_EXISTSu_DEF, aMODu_MULTu_ADD, aDISTu_ADD2, aSUMu_LT, aADMISSIBLEu_BASE, anumu_INDUCTION, aNUMu_REPu_RULESu_conjunct0, aELu_APPEND, adot, aSUMu_NEG, anaddu_rep, aCARDu_SUBSETu_EQ, aHREALu_OFu_NUMu_MUL, aCARDu_LEu_POWERSET, aTREALu_MULu_ASSOC, aEQu_C, aINTu_MULu_LZERO, aSeq2o_TL, aCARTu_EQu_FULL, aBIJECTIVEu_LEFTu_RIGHTu_INVERSE, aINTu_LEu_NEGTOTAL, aREALu_EQu_SUBu_LADD, aABSORPTION, aWFu_SUBSET, aNADDu_LEu_WELLDEFu_LEMMA, ainf, aARITHu_SUBu_conjunct9, aprodu_tybiju_conjunct1, aARITHu_GTu_conjunct3, aDISTu_TRIANGLEu_HALFu_R, aREALu_NEGu_0, aCARDu_FUNSPACEu_LE, aINSEGu_PROPERu_SUBSETu_FL, aDOTu_3, aCONSTR, aREALu_LTu_SQUAREu_ABS, aMULTu_SUC, aCARDu_CLAUSESu_conjunct0, aCARDu_UNIONu_OVERLAP, aBIJECTIONSu_HASu_SIZEu_EQ, aCARDu_UNIONu_OVERLAPu_EQ, aREALu_INVu_0, aINFINITEu_FROM, aARITHu_EQu_conjunct6, aLEFTu_IMPu_EXISTSu_THM, aORTHOGONALu_BASISu_BASIS, aINTu_LEu_NEGR, aTRIVu_EXISTSu_ANDu_THM, aREALu_LEu_SUBu_LADD, aCOUNTABLEu_LIST, aPSUBSETu_INSERTu_SUBSET, aFINITEu_EMPTY, aREALu_WLOGu_LE, aNOTu_UNIVu_PSUBSET, aLENGTHu_EQu_CONS, aINFu_UNIQUEu_FINITE, aNUMSEGu_CLAUSESu_conjunct1, aSUMu_POSu_LEu_NUMSEG, aSUMu_ABS, aWOSETu_TOTALu_LT, aINTu_LTu_DISCRETE, aINTu_ADDu_LID, aINu_ELIMu_THMu_conjunct0, aSUMu_EQ, aVECTORu_MULu_LCANCEL, aREALu_LETu_ADD, aFINITEu_FINITEu_PREIMAGE, ahrealu_add, aREALu_EQu_ADDu_LCANCELu_0, aARITHu_EXPu_conjunct8, aINTu_POWu_POW, aHASu_SIZEu_SUC, aREALu_ARCHu_LT, ahrealu_ofu_num, aCARDu_EQu_CARD, aREALu_ADDu_LINV, aBIJECTIVEu_ONu_LEFTu_RIGHTu_INVERSE, aEXISTSu_INu_GSPECu_conjunct0, aINTu_NEGu_MINUS1, aREALu_LTu_RMULu_EQ, aINTu_ADDu_LINV, aMATRIXu_CMULu_LID, aINTu_SUBu_TRIANGLE, aSQRTu_MONOu_LE, aREALu_LTu_LSQRT, aSING, aHASu_SIZEu_FINITEu_IMAGE, aFINITEu_REALu_INTERVALu_conjunct2, aINDu_SUCu_0u_EXISTS, aisu_nadd, aDIVMODu_EXIST, aINTu_MULu_ACu_conjunct2, aLASTu_APPEND, aARITHu_LEu_conjunct1, aFINITEu_PRODUCT, aUNWINDu_THM1, aintu_tybiju_conjunct1, anumu_RECURSION, aREALu_POWu_EQu_1, aWFu_DCHAIN, aCARDu_LEu_EQu_SUBSET, aNADDu_MULu_ASSOC, aINTu_MINu_ACIu_conjunct2, alistu_CASES, aUNIONu_FL, aINTu_NEGu_EQ, aREALu_LTu_TOTAL, aCHAINu_SUBSET, aREALu_ABSu_NZ, aINTu_MAXu_ACIu_conjunct1, aHREALu_OFu_NUMu_LE, aARITHu_ADDu_conjunct1, aNSUMu_PAIR, aVSUMu_UNIONu_LZERO, aINu_REST, aIMAGEu_ID, atoset, aNADDu_MULu_WELLDEF, aMAPu_FSTu_ZIP, aBILINEARu_VSUM, aUNIONu_ACIu_conjunct0, aITLIST2u_conjunct1, aintu_le, aEQu_ADDu_LCANCEL, aMEASURE, aEXISTSu_UNIQUEu_REFL, aMATRIXu_ADDu_ACu_conjunct0, aINTu_LTu_SQUAREu_ABS, aMODu_1, aNSUMu_OFFSET, aVSUMu_CONST, aINTu_SGNu_NEG, aINu_ELIMu_THMu_conjunct1, aREALu_LTu_MIN, aHASu_SIZEu_INDEX, aFINITEu_UNION, aINSERTu_DIFF, aCARDu_DIFF, aREALu_LEu_RINV, aREALu_ABSu_SUPu_LE, aREALu_ABSu_BETWEEN2, aINTu_POWu_ZERO, aINTu_NOTu_EQ, aCARDu_UNIONu_LE, aLTu_MULTu_RCANCEL, aNORMu_LE, aINTu_LETu_TOTAL, aINFINITYu_AX, aBILINEARu_RADD, alinear, aITERATEu_UNIONu_GEN, aREALu_SUBu_RZERO, aEXPu_EQu_0, aTREALu_ADDu_ASSOC, aINTu_MINu_SYM, aWFu_IND, aREALu_INFu_LEu_FINITE, aREALu_SUBu_LDISTRIB, aUNIONSu_SUBSET, aBILINEARu_LZERO, arealu_mul, aintu_neg, aMATRIXu_VECTORu_MULu_LINEAR, aSUMu_BOUNDu_LTu_ALL, aCARDu_LEu_COUNTABLEu_SUBSETS, aINTu_MAXu_ASSOC, aREALu_POWu_MONOu_INV, aNEUTRALu_ADD, aCONDu_EXPAND, aINTu_POWu_conjunct0, aVSUMu_SUC, aSETu_OFu_LISTu_APPEND, aARITHu_SUCu_conjunct3, aWLOGu_LT, aCARDu_EQu_EMPTY, aORTHOGONALu_CLAUSESu_conjunct3, aEXPu_conjunct0, aNSUMu_SUPERSET, atrealu_ofu_num, aLINEARu_COMPOSEu_ADD, aFORALLu_INu_GSPECu_conjunct1, atrealu_mul, aNORMu_TRIANGLEu_LT, aARITHu_MULTu_conjunct7, aSUBSETu_INSERT, aMULTu_ASSOC, aALLu_conjunct0, achain, aMATRIXu_INV, aINTu_LEu_NEGL, aVECTORu_NEGu_EQu_0, aNSUMu_0, afstcart, aINTu_GCDu_EXISTSu_POS, aEXCLUDEDu_MIDDLE, aEXPu_ADD, aVECTORu_ADDu_RDISTRIB, anaddu_add, aSUMu_POSu_EQu_0, aMATRIXu_NEGu_EQu_0, aFORALLu_INu_UNIONS, aEXISTSu_SUBSETu_IMAGE, aREALu_RSQRTu_LE, aDIVu_EQu_EXCLUSION, aTREALu_MULu_LID, aSUMu_EQu_GENERALu_INVERSES, aHASu_SIZEu_UNION, aSUMu_EQu_0, aCARDu_LEu_CARDu_IMP, aSUMu_SINGu_NUMSEG, aINTu_LTEu_TRANS, aREALu_LNEGu_UNIQ, aARITHu_LTu_conjunct3, aFINITEu_SUBSET, aFINITEu_NUMSEGu_LT, aHULLu_Pu_ANDu_Q, aINTu_LEu_LADD, aFORALLu_SUC, aINTu_LTu_LMULu_EQ, aITERATEu_EQu_GENERALu_INVERSES, anumu_CASES, aCOMPONENT, aSUBSETu_INTERu_ABSORPTION, aWFu_INTu_MEASURE, aHREALu_LEu_REFL, aWFu_FINITE, aREALu_SUBu_RDISTRIB, aLENGTHu_REPLICATE, aVECTORu_MATRIXu_MULu_TRANSP, aUNIONSu_0, aREALu_POWu_EQu_ODDu_EQ, aREALu_MAXu_LT, aNSUMu_BOUNDu_GEN, aREALu_LTu_INV2, aMATRIXu_MULu_LZERO, aVSUMu_ADDu_NUMSEG, aLINEARu_COMPOSEu_NEG, aREALu_DIVu_1, aMEMu_ASSOC, aEQu_ADDu_LCANCELu_0, aMAPu_EQ, aITERATEu_UNION, aDOTu_BASIS, aDISTu_EQu_0, aREALu_EQu_ADDu_RCANCEL, aVSUMu_NEG, aINTu_MULu_ASSOC, aREALu_LTu_MULu_EQu_conjunct1, aDELETEu_DELETE, aREALu_MAXu_LE, aREALu_OFu_NUMu_POW, aSURJECTIVEu_IMAGEu_THM, aREALu_LTu_RDIVu_EQ, aou_DEF, aREALu_OFu_NUMu_SUC, aAPPENDu_ASSOC, aITERATEu_EQu_GENERAL, aALL2u_conjunct2, aNUMu_REPu_CASES, aVSUMu_0, aREALu_INVu_MUL, aREALu_POWu_NZ, aLTEu_TRANS, aNSUMu_CLOSED, aCASEWISEu_CASES, aEXPu_EQu_1, aNADDu_OFu_NUMu_MUL, aINTu_WOP, aDIVISIONu_SIMPu_conjunct1, aCARDu_LTu_TOTAL, aADJOINTu_LINEAR, aVSUMu_SING, aSKOLEMu_THM, aPAIRu_SURJECTIVE, aCOUNTABLEu_EMPTY, aINTu_OFu_NUMu_MIN, aHREALu_LEu_ADD2, aVSUMu_SUBu_NUMSEG, afiniteu_index, aHASu_SIZEu_1, aAPPENDu_conjunct1, aMAP2u_DEFu_conjunct1, aINTu_EQu_NEG2, aMULTu_EQu_1, aZCONSTR, aCOUNTABLEu_ASu_IMAGEu_SUBSET, aITERATEu_EQu_NEUTRAL, aINTu_POWu_EQ, aREALu_SUBu_ADD, aREALu_POWu_LT, aINTu_SUBu_SUB, aREALu_LTu_LE, aSUMu_LE, aprodu_tybiju_conjunct0, aARITHu_SUBu_conjunct0, aFINITEu_POWERSET, aCOUNTABLEu_UNIONS, aCARDu_LEu_TOTAL, aARITHu_LEu_conjunct7, aREPLICATEu_conjunct0, aREALu_POWu_EQu_ODD, aVECu_COMPONENT, aNADDu_LEu_ANTISYM, aVECTORu_MULu_EQu_0, aARITHu_EXPu_conjunct7, aAPPENDu_BUTLASTu_LAST, aREALu_ARCH, aVSUMu_DELETE, aREALu_POWu_ADD, aLINEARu_NEGATION, aVECTORu_NEGu_COMPONENT, aDOTu_POSu_LE, aFINITEu_SUPPORTu_DELTA, aREALu_LEu_TRANS, aNSUMu_MULTICOUNTu_GEN, aMATRIXu_CMULu_SUBu_RDISTRIB, aVSUMu_TRIVu_NUMSEG, aVSUMu_CLAUSESu_NUMSEGu_conjunct1, aVECTORu_ADDu_COMPONENT, aINTu_BOUNDSu_LT, aVECu_EQ, aINu_DELETE, aCARDu_EQu_CART, aREALu_SUPu_UNIQUE, aFINITEu_FINITEu_IMAGE, aNSUMu_SUPPORT, aFLu_SUC, aVSUMu_NORM, aNSUMu_ADDu_SPLIT, acartu_tybiju_conjunct0, ansum, aLEu_CASES, aINTu_LEu_SUBu_LADD, aISOu_FUN, aBASISu_INJ, asetu_ofu_listu_conjunct0, aMATRIXu_SUB, aCARDu_EQu_CARDu_IMP, aREALu_MAXu_ASSOC, aCOUNTABLEu_FINITEu_SUBSETS, aFCONSu_conjunct1, aPAIRu_EXISTSu_THM, aEXISTSu_VECTORu_3, aFINITEu_REALu_INTERVALu_conjunct6, aODDu_EXISTS, aINTu_LTu_MAX, awoset, aNORMu_POSu_LT, aREALu_MAXu_ACIu_conjunct3, aVSUMu_VSUMu_PRODUCT, aREALu_LEu_SUBu_RADD, aCARDu_LEu_CONG, aINTu_LEu_ADDR, aNADDu_MULu_LINVu_LEMMA2, aODDu_MULT, aARITHu_LTu_conjunct7, aNUMu_COUNTABLE, aMODu_ADDu_MOD, aBOUNDSu_DIVIDED, aNSUMu_EQu_GENERALu_INVERSES, aODDu_EXP, aANDu_DEF, aREALu_ABSu_TRIANGLE, aREALu_HALFu_conjunct0, aLEFTu_ADDu_DISTRIB, aSUMu_PARTIALu_SUC, aREALu_LTu_01, aINTu_POWu_MUL, aORTHOGONALu_CLAUSESu_conjunct0, aCOUNTABLEu_IMAGE, aHREALu_EQu_ADDu_RCANCEL, aREALu_MINu_MIN, aSUPERADMISSIBLEu_CONST, aMATRIXu_ADDu_RID, aARITHu_MULTu_conjunct3, aSUMu_ADDu_SPLIT, aARITHu_ADDu_conjunct9, aDOTu_RSUB, aREALu_SGNu_0, aMATRIXu_EQUALu_COLUMNS, aARITHu_SUBu_conjunct4, aINTu_ADDu_RINV, alistu_RECURSION, aoneu_Axiom, au_UNGUARDEDu_PATTERN, aDISTu_TRIANGLEu_LT, aFINITEu_RECURSIONu_DELETE, aCOUNTABLEu_DIFFu_FINITE, aARITHu_EQu_conjunct0, aZCONSTRu_ZBOT, aINTu_LTu_IMPu_LE, aINSEGu_LINSEG, aHREALu_INVu_0, aLINEARu_ADD, aREPu_ABSu_PAIR, aINTu_LTu_ADDNEG2, aDOTu_1, aREALu_SUBu_SUB, aNSUMu_IMAGEu_NONZERO, aDOTu_NORMu_NEG, aUNIQUEu_SKOLEMu_ALT, arealu_abs, aALL2u_ANDu_RIGHT, aSUMu_SUMu_PRODUCT, aNSUMu_INJECTION, aFORALLu_3, aREALu_MULu_POSu_LT, aREALu_LEu_INFu_SUBSET, aARITHu_ODDu_conjunct2, aSUBSETu_UNIONS, aNSUMu_OFFSETu_0, aNORMu_TRIANGLEu_LE, aISO, aREALu_ABSu_0, aORTHOGONALu_RNEG, aREALu_MULu_LINV, aITERATEu_INJECTION, aINu_CARDu_MUL, aINTu_SGNu_EQu_conjunct2, aREALu_LTu_LINV, aMAPu_EQu_NIL, aBILINEARu_RMUL, arealu_add, aUNION, aINTu_LEu_POW2, aREALu_LEu_RSQRT, aCARDu_HASu_SIZEu_CONG, aREALu_LTu_RADD, aFINITEu_FINITEu_PREIMAGEu_GENERAL, aREALu_MULu_RINVu_UNIQ, aREALu_ABSu_STILLNZ, aINTu_POWu_LT2u_REV, aBIT0, arealu_lt, aOUTR, aWFu_EQ, aITERATEu_PAIR, aINTu_EXISTSu_ABS, aMATRIXu_VECTORu_MULu_RMUL, aDOTu_LMULu_MATRIX, aARITHu_EQu_conjunct3, aALL2u_DEFu_conjunct0, aNADDu_DIST, avectoru_norm, aCARTu_EQ, aREALu_LTu_POW2, aREALu_NEGu_EQ, aREALu_SUBu_0, aINTu_LTu_REFL, aINSEGu_PROPERu_SUBSET, aBOOLu_CASESu_AX, aBOUNDSu_IGNORE, aWFu_LEX, aFINITEu_INDUCT, aMAPu_conjunct1, aFINITEu_REALu_INTERVALu_conjunct3, aCARDu_SUBSETu_IMAGE, aINTERu_ACIu_conjunct1, alistu_ofu_set, aNORMu_CAUCHYu_SCHWARZ, aNSUMu_SING, aEXISTSu_SIMP, aINTu_POWu_LE2u_REV, aREALu_LEu_MUL2, aNADDu_ALTMUL, aFORALLu_NOTu_THM, aREALu_MULu_RID, aCARDu_NUMSEGu_1, arealu_max, aWFu_RECu_TAILu_GENERALi_, aNOTu_CLAUSESu_conjunct0, aREALu_ADDu_ACu_conjunct2, aisu_naddu_0, aINSERTu_UNIONu_EQ, amatrixu_neg, aMONOu_AND, aNSUMu_GROUP, aARITHu_GEu_conjunct2, aINTu_LTu_ADDL, aNORMu_LEu_L1, aBILINEARu_LADD, aFORALLu_UNWINDu_THM1, aSUMu_EQu_GENERAL, aMONOu_NOT, aVECTORu_CHOOSEu_DIST, aCONJu_ASSOC, aUNIONS, aRIGHTu_IMPu_FORALLu_THM, aCARDu_EQu_REAL, aNOTu_EVEN, aREALu_MINu_MAX, aNSUMu_CLAUSESu_LEFT, aREALu_INFu_ASCLOSE, aIMPu_DEF, arealu_invu_th, aSURJ, aNOTu_ODD, aFORALLu_FINITEu_SUBSETu_IMAGE, aLEu_0, aMATRIXu_MULu_VSUM, aDIMINDEXu_FINITEu_SUM, aNOTu_CONSu_NIL, aINTu_MINu_MAX, aFLATTENu_LEMMA, aMINIMAL, aDISTu_EQ, aNSUMu_EQu_0u_NUMSEG, aNORMu_EQu_1, aEXISTSu_INu_CLAUSESu_conjunct0, aREALu_LETu_TOTAL, aREVERSEu_conjunct1, aintu_sgn, aORDINALu_SUC, aREALu_LTu_LMULu_EQ, aCARDu_EQu_BIJECTION, aINJNu_INJ, aDIFF, aSUBSETu_DIFF, aSUPPORTu_CLAUSESu_conjunct4, aREALu_INVu_INV, aINTu_OFu_NUMu_LT
% 198.40/131.64  % No. of inferences in proof: 6356
% 198.40/131.64  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 131107 ms resp. 129981 ms w/o parsing
% 209.20/137.16  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 209.31/137.23  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------